Chapter 10
Connections with totally skew-symmetric torsion and
nearly-Kähler geometry
Paul-Andi Nagy
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . .
Almost Hermitian geometry . . . . . . . . . . . . . .
2.1 Curvature tensors in the almost Hermitian setting
2.2 The Nijenhuis tensor and Hermitian connections .
2.3 Admissible totally skew-symmetric torsion . . .
2.4 Hermitian Killing forms . . . . . . . . . . . . .
3 The torsion within the G1 class . . . . . . . . . . . . .
3.1 G1 -structures . . . . . . . . . . . . . . . . . . .
3.2 The class W1 C W4 . . . . . . . . . . . . . . . .
3.3 The u.m/-decomposition of curvature . . . . . .
4 Nearly-Kähler geometry . . . . . . . . . . . . . . . .
4.1 The irreducible case . . . . . . . . . . . . . . . .
5 When the holonomy is reducible . . . . . . . . . . . .
5.1 Nearly-Kähler holonomy systems . . . . . . . .
5.2 The structure of the form C . . . . . . . . . . .
5.3 Reduction to a Riemannian holonomy system . .
5.4 Metric properties . . . . . . . . . . . . . . . . .
5.5 The final classification . . . . . . . . . . . . . .
6 Concluding remarks . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
4
6
11
14
16
22
22
25
26
31
35
36
36
38
41
44
46
48
49
1 Introduction
For a given Riemannian manifold .M n ; g/, the holonomy group Hol.M; g/ of the
metric g contains fundamental geometric information. If the manifold is locally irreducible, as it can always be assumed locally by the well-known de Rham splitting
theorem, requiring that Hol.M; g/ is smaller that SO.n/ has strong geometric implications at the level of the geometric structure supported by M (see [52] for an account
of the classification of irreducible Riemannian holonomies). Nowadays, this has been
extended to cover the classification of possible holonomy groups of torsion-free affine
connections (see [9], [10], [44] for an overview).
2
connexion
or
connection??
Paul-Andi Nagy
From a somewhat different perspective, in the past years there has been increased
interest in the theory of metric connections with totally skew-symmetric torsion on
Riemannian manifolds. The holonomy group of such a connexion is once again an
insightful object which can be used to describe the geometry in many situations. One
general reason supporting this fact is that the de Rham splitting theorem fails to hold
in this generalised setting and the counterexamples to this extent are more intricate
than just Riemannian products. Additional motivation for the study of connections in
the above mentioned class comes from string theory, where this requirement is part of
various models (see [1], [54] for instance).
Actually, some of these directions go back to the work of A.Gray, in the seventies,
in connection to the notion of weak holonomy [31], [3], aimed to generalise that of
Riemannian holonomy. The study of nearly-Kähler manifolds initiated by Gray in
[30], [32], [33] has been instrumental in the theory of geometric structures of nonintegrable type. As it is well known, for almost Hermitian structures a classification
has been first given by Gray and Hervella in [34], which proved to be quite suitable
for generalisation.
Recently, general G-structures of non-integrable type have been studied in [55],
[26], [17], [25], by taking into account the algebraic properties of the various torsion
modules. Using results from holonomy theory they have indicated how to read off
various intersections of G-modules interesting geometric properties when in presence,
say, of a connection with totally skew-symmetric torsion. Aspects of such an approach
are contained in the Gray–Hervella classification and its analogues for other classical
groups, we shall comment on later. Note that a central place in [17] is occupied by
connections with parallel skew-symmetric torsion, studied since then by many authors.
By combining these two points of view, let us return to the Gray–Hervella classification and recall that one of its subclasses, that of nearly-Kähler structures, has a
priori parallel torsion for a connection with totally skew-symmetric torsion [40]. This
suggests perhaps that geometric properties of the intrinsic torsion might by obtained
in more general context.
In this chapter we will at first investigate some of the properties of almost Hermitian
structures in the Gray–Hervella class W1 C W3 C W4 , also referred to – as we shall in
what follows – as the class G1 . It is best described [26] as the class of almost Hermitian
structures admitting a metric and Hermitian connection with totally skew-symmetric
torsion. Yet, this property is equivalent [26] with requiring the Nijenhuis tensor of the
almost complex structure to be a 3-form when evaluated using the Riemannian metric
of the structure. There are several subclasses of interest, for example the class W3 CW4
consisting of Hermitian structures, the class W4 of locally conformally Kähler metrics
and the class W1 of nearly-Kähler structures, which will occupy us in the second half of
the exposition. One general question which arises is how far the class W1 C W3 C W4
is, for instance, from the subclass W3 C W4 of Hermitian structures.
The chapter is organised as follows. In Section 2 we present a number of algebraic
facts related to representations of the unitary group and also some background material
on algebraic curvature tensors. We also briefly review some basic facts from almost-
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
3
Hermitian geometry, including a short presentation of the Hermitian connections of
relevance for what follows. At the end of Section 2 we introduce the notion of Hermitian Killing form with respect to a Hermitian connection with torsion. Although this
is a straightforward variation on the Riemannian notion it will serve an explanatory
rôle in the next section.
Section 3 is devoted to the study of the properties of the torsion tensor of an almost
Hermitian structure in the class G1 . More precisely we prove:
Theorem 1.1. Let .M 2m ; g; J / be an almost-Hermitian structure of type G1 and let
D be the unique Hermitian connection whose torsion tensor T D belongs to ƒ3 .M /.
Then
1
DX C D .X ³ d T D /3
2
C
for all X in TM , where
is proportional to the Nijenhuis form of J and the subscript
indicated orthogonal projection on 3 .M /, the bundle of real valued forms of type
.0; 3/ C .3; 0/.
For unexplained notation and the various numerical conventions we refer the reader
to Section 2. In the language of that section, this can be rephrased to say that C
is a Hermitian Killing form of type .3; 0/ C .0; 3/. Theorem 1.1 is extending to the
wider class of G1 -structures the well-known [40] parallelism of the Nijenhuis tensor
of a nearly-Kähler structure. We also give necessary and sufficient conditions for the
parallelism of the Nijenhuis tensor, in a G1 -context and specialise Theorem 1.1 to the
subclass of W1 C W4 manifolds which has been recently subject of attention [13],
[16], [47] in dimension 6. Section 3 ends with the explicit computation of the most
relevant components of the curvature tensor of the connection D above, in a ready to
use form.
Section 4 contains a survey of available classification results in nearly-Kähler
geometry. Although this is based on [48], [49] we have adopted a slightly different
approach, which emphasises the rôle of the non-Riemannian holonomy system which
actually governs the way various holonomy reductions are performed geometrically.
We have outlined, step by step, the procedure reducing a holonomy system of nearlyKähler type to an embedded irreducible Riemannian holonomy system, which is the
key argument in the classification result which is stated below.
Theorem 1.2 ([49]). Let .M 2m ; g; J / be a complete SNK-manifold. Then M is, up
to finite cover, a Riemannian product whose factors belong to the following classes:
(i) homogeneous SNK-manifolds;
(ii) twistor spaces over positive quaternionic Kähler manifolds;
(iii) 6-dimensional SNK-manifolds.
We have attempted to make the text as self-contained as possible, for the convenience of the reader.
runningheads
for the title
shortened,
please
check
4
Paul-Andi Nagy
2 Almost Hermitian geometry
An almost Hermitian structure .g; J / on a manifold M consists in a Riemannian metric
g on M and a compatible almost complex structure J . That is, J is an endomorphism
of the tangent bundle to M such that J 2 D 1TM and
g.JX; J Y / D g.X; Y /
for all X, Y in TM . It follows that the dimension of M is even, to be denoted by 2m
in the subsequent. Any almost Hermitian structure comes equipped with its so-called
Kähler form ! D g.J ; / in ƒ2 . Since ! is everywhere non-degenerate the manifold
M is naturally oriented by the top degree form ! m .
We shall give now a short account on some of the U.m/-modules of relevance
for our study. Let ƒ? denote the space of differential p-forms on M , to be assumed
real-valued, unless stated otherwise.
We consider the operator J W ƒp ! ƒp acting on a p-form ˛ by
.J˛/.X1 ; : : : ; Xp / D
p
X
˛.X1 ; : : : ; JXk ; : : : ; Xp /
kD1
for all X1 ; : : : ; Xp in TM . For future use let us note that alternatively
J˛ D
2m
X
ei[ ^ .Jei ³ ˛/
iD1
for all ˛ in ƒ? , where fei ; 1 i 2mg is some local orthonormal frame on M . The
almost complex structure J can also be extended to ƒ? by setting
.J˛/.X1 ; : : : ; Xp / D ˛.JX1 ; : : : ; JXp /
for all Xi , 1 i p in TM .
Let us recall that the musical isomorphism identifying vectors and 1-forms is given
by X 2 TM 7! X [ D g.X; / 2 ƒ1 . Note that in our present conventions one has
JX [ D .JX/[ whenever X belongs to TM .
Now J acts as a derivation on ƒ? and gives the complex bi-grading of the exterior
algebra in the following sense. Let p;q be given as the .p q/2 -eigenspace of J 2 .
One has an orthogonal, direct sum decomposition
M
p;q ;
ƒs D
pCqDs
called the bidegree splitting of ƒs , 0 s 2m. Note that p;q D q;p . Of special
importance in our discussion are the spaces p D p;0 ; forms ˛ in p are such that
the assignment
.X1 ; : : : ; Xp / ! ˛.JX1 ; X2 ; : : : ; Xp /
5
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
is still an alternating form which equals p 1 J˛. We now consider the operator
L W ƒ? ! ƒ? given as multiplication with the Kähler form !, together with its adjoint,
to be denoted by L? . Forms in
ƒ?0 D Ker L?
are called primitive and one has
ŒL? ; L D .m p/1ƒp
on the space of p-forms. Note that the inclusion p ƒp0 holds whenever p 0.
Let p ˝1 q be the space of tensors Q W p ! q which satisfy
QJ D J Q;
where J W p ! p is given by J D p 1 J. We also define p ˝2 q as the space of
tensors Q W p ! q such that QJ D J Q. The following lemma is easy to verify.
Lemma 2.1. Let a W p ˝ q ! ƒpCq be the total alternation map. Then:
(i) the image of the restriction of a to p ˝1 q ! ƒpCq is contained in p;q ;
(ii) the image of the restriction of a to p ˝2 q ! ƒpCq is contained in pCq ;
(iii) the total alternation map a W p ˝1 q ! ƒpCq is injective for p ¤ q;
(iv) the kernel of a W p ˝ q ! ƒpCq is contained in p ˝2 q .
In this chapter most of our computations will involve forms of degree up to 4. For
further use we recall that one has
JDJ
on 1;2 and also that
J2;2 ˚4 D 12;2 ˚4 ; J1;3 D 11;3 :
We will now briefly introduce a number of algebraic operators which play an important
rôle in the study of connections with totally skew-symmetric torsion. If ˛ belongs to
ƒ2 and ' is in ƒ? we define the commutator 1
Œ˛; ' D
2m
X
.ei ³ ˛/ ^ .ei ³ '/;
iD1
where fei ; 1 i 2mg is a local orthonormal frame in TM . Note that
Œ!; ' D J'
for all ' in ƒ? . It is also useful to mention that
Œ1;1 ; 3 3 ;
1 This
Œ1;1 ; 1;2 1;2 ;
Œ2 ; 3 1;2 :
is proportional to the commutator in the Clifford algebra bundle Cl.M / of M .
(2.1)
6
Paul-Andi Nagy
Lastly, for any couple of forms .'1 ; '2 / in ƒp ƒq we define their product '1 • '2
in ƒpCq2 by
2m
X
.ek ³ '1 / ^ .ek ³ '2 /;
' 1 • '2 D
kD1
for some local orthonormal frame fek ; 1 k 2mg on M . We end this section
by listing a few properties of the product • for low degree forms, to be used in what
follows.
Lemma 2.2. The following hold:
(i) '1 • '2 belongs to 2;2 ˚ 1;3 for all '1 ; '2 in 1;2 ;
(ii) ' • J' is in 1;3 for all ' in 1;2 ;
(iii) ' •
(iv)
(v) if
1•
belongs to 1;3 ˚ 4 whenever ' is in 1;2 and all
2
belongs to 2;2 for all
is in 3 we have that
1;
2
in 3 ;
in 3 ;
D 0.
•J
The proofs consist in a simple verification which is left to the reader.
2.1 Curvature tensors in the almost Hermitian setting
We shall present here a number of basic facts concerning algebraic curvature tensors
we shall need later on. Since this is intended to be mainly at an algebraic level, our
context will be that of a given Hermitian vector space .V 2m ; g; J /. Let us recall that
the Bianchi map b1 W ƒ2 ˝ ƒ2 ! ƒ1 ˝ ƒ3 is defined by
.b1 R/x D
2m
X
ei[ ^ R.ei ; x/
iD1
for all R in ƒ ˝ ƒ and for all x in V . The space of algebraic curvature tensors on
V is given by
K.so.2m// D .ƒ2 ˝ ƒ2 / \ Ker.b1 /
2
2
and it is worth observing that K.so.2m// D S 2 .ƒ2 / \ Ker.a/. Restricting the group
to the unitary one makes appear the space of Kähler curvature tensors given by
K.u.m// D .1;1 ˝ 1;1 / \ Ker.b1 / D S 2 .1;1 / \ Ker.a/:
We shall briefly present some well-known facts related to the space of Kähler curvature
tensors, with proofs given mostly for the sake of completeness. First of all, we have
an inclusion of 2;2 into S 2 .1;1 / given by
y
7! ;
7
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
y
where .x;
y/ D 14 ..x; y/ C .J x; Jy// for all x, y in V . This is mainly due to
the fact that forms of type .2; 2/ are J -invariant. A short computation yields
y D
b1 ./
(2.2)
{
for all in 2;2 . There is also an embedding of 1;3 into 2 ˝ 1;1 given by 7! ,
where
{
.x;
y/ D .x; y/ .J x; Jy/
for all x, y in V . It is easily verified that
{ x D J x ³ J 2x ³ .b1 /
(2.3)
for all x, y in V and for all in 1;3 .
Similarly to the well-known splitting S 2 .ƒ2 / D K.so.2m// ˚ ƒ4 we have:
Proposition 2.1. There is an orthogonal, direct sum splitting
S 2 .1;1 / D K.u.m// ˚ 2;2 :
y for some Kähler
Explicitly, any Q in S 2 .1;1 / can be uniquely written as Q D R C curvature tensor R and some in 2;2 .
Proof. Let Q belong to S 2 .1;1 /. It satisfies Q.J x; Jy/ D Q.x; y/ for all x, y in V
and since Q belongs to S 2 .ƒ2 / it splits as
Q D R C ;
where R is in K.so.2m// and belongs to ƒ4 . In particular
R.J x; Jy/ C .J x; Jy/ D R.x; y/ C .x; y/
(2.4)
for all x, y in V .
Since Q belongs to 1;1 ˝ 1;1 we have that .b1 Q/x belongs to 1;2 for all x in V .
But R is a curvature tensor, hence .b1 Q/x D 3x ³ , leading to x ³ in 1;2 for
all x in V . It is then easy to see that must be an element of 2;2 . It follows by direct
verification that R given by
1
R .x; y/ D .J x; Jy/ .x; y/
3
belongs to K.so.2m//. Setting now R0 D R 34 R it is easy to see from (2.4) that
R0 belongs to K.u.m// and our claim follows after appropriate rescaling from
Q.x; y/ D R0 .x; y/ C
for all x, y in V .
3
.J x; Jy/ C .x; y/
4
Our next and last goal in this section is to have an explicit splitting of certain
elements of ƒ2 ˝ 1;1 along ƒ2 D 1;1 ˚ 2 . More explicitly, we will look at this in
8
Paul-Andi Nagy
the special case of a tensor R in ƒ2 ˝ 1;1 such that
R.x; y; z; u/ R.z; u; x; y/ D x .y; z; u/ y .x; z; u/
z .u; x; y/ C u .z; x; y/
(2.5)
holds for all x, y, z, u in V . Here belongs to ƒ1 ˝ ƒ3 and in what follows we shall
use the notation
T D a. /
for the total alternation of . The tensor R is the algebraic model for the curvature
tensor of a Hermitian connection with totally skew-symmetric torsion, which will be
our object of study later on in the chapter.
Remark 2.1. A complete decomposition of K.so.2m// into irreducible components
under the action of U.m/ has been given by Tricerri and Vanhecke in [56]. Further
information concerning the splitting of ƒ2 ˝ ƒ2 , again under the action of U.m/ is
given in detail in [20]. While the material we shall present next can be equivalently
derived from these references, it is given both for self-containedness and also as an
illustration that one can directly proceed, in the case of a connexion with torsion, to
directly split its curvature tensor without reference to the Riemann one. As a general
observation we also note that the procedure involves only control of the orthogonal
projections onto the relevant U.m/-submodules.
Recall that for any ˛ in ƒ2 its orthogonal projection on 1;1 is given by
1
.˛ C J˛/:
2
To obtain the decomposition of R we need the following preliminary lemma.
˛1;1 D
Lemma 2.3. Let in 1 ˝1;2 be given. We consider the tensors H1 , H2 in ƒ2 ˝1;1
given by
H1 .x; y/ D .x ³ y y ³ x /1;1 and H2 .x; y/ D H2 .J x; Jy/
for all x, y in V . Then:
(i) 2.b1 H1 /x D 3x C J J x C x ³ a./ J x ³ a.J /;
(ii) 2.b1 H2 /x D x C 3J J x C J x ³ ac ./ C x ³ ac .J /
for all x in V . Here J in 1 ˝ 1;2 is defined by .J /x D J x for all x in V , and
the complex alternation map ac W 1 ˝ 1;2 ! ƒ4 is given by
a ./ D
c
2m
X
kD1
ek[ ^ Jek :
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
9
Proof. (i) Let fek ; 1 k 2mg be an orthonormal basis in V . Directly from its
definition, the tensor H1 is given by
2H1 .x; y/ D x ³ y J x ³ J y y ³ x C Jy ³ J x ;
hence
2.b1 H1 /x D 4x 2m
X
ek[ ^ .x ³ ek / C
kD1
since J D J on 1;2
2m
X
ek[ ^ .J x ³ J ek /
kD1
. Now
x ³ a./ D x 2m
X
ek[ ^ .x ³ ek /
kD1
and
J x ³ a.J / D J J x 2m
X
ek[ ^ .J x ³ J ek /
kD1
for all x in V and the claim follows immediately.
(ii) It is enough to use (i) when replacing by J .
We split now
D 1;2 C 3
along ƒ1 ˝ ƒ3 D .ƒ1 ˝ 1;2 / ˚ .ƒ1 ˝ 3 / and also
T D T 2;2 C T 1;3 C T 4
along the bidegree decomposition of ƒ4 in order to be able to introduce some of the
components of the tensor R. These are the tensor Ra in 1;1 ˝ 1;1 given by
1
Ra .x; y/ D .x yy x/1;1 C.J x JyJy J x/1;1 .T 2;2 .x; y/CT 2;2 .J x; Jy//
2
for all x, y in V , and the tensor Rm in 2 ˝ 1;1 defined by
1
Rm .x; y/ D .x yy x/1;1 .J x JyJy J x/1;1 .T 1;3 .x; y/T 1;3 .J x; Jy//
2
for all x, y in V . Here we use x y as a shorthand for y ³ x , whenever x, y are in V .
The promised decomposition result for R is achieved mainly by projection of (3.5)
onto
ƒ2 ˝ 1;1 D .1;1 ˝ 1;1 / ˚ .1;1 ˝ 2 /
while taking into account that R belongs to ƒ2 ˝ 1;1 .
Theorem 2.1. Let R belong to ƒ2 ˝ 1;1 such that (2.5) is satisfied. We have a
decomposition
y C 1 Ra C Rm ;
R D RK C 2
“when” ok?
10
Paul-Andi Nagy
where RK belongs to K.u.m// and is in 2;2 . Moreover:
(i) Ra belongs to ƒ2 .1;1 / and satisfies the Bianchi identity
1
1
.b1 Ra /x D 2.x1;2 C J J1;2
x / x ³ A1 J x ³ A2
2
2
for all x in V . The forms A1 and A2 are explicitly given by
A1 D a. 1;2 / C ac .J 1;2 / 4T 2;2 ;
A2 D ac . 1;2 / a.J 1;2 /:
(ii) The Bianchi identity for Rm in 2 ˝ 1;1 is
1
1
.b1 Rm /x D .x1;2 J J1;2
x / x ³ A3 C J x ³ A4
2
2
for all x in V , where
A3 D a. 1;2 / ac .J 1;2 / 2T 1;3 ;
A4 D a.J 1;2 / C ac . 1;2 / JT 1;3 :
Proof. We first show that Ra belongs to ƒ2 .1;1 /. Directly from the definition of the
map a we have that the tensor
1
Q.x; y/ WD x y y x T .x; y/
2
belongs to ƒ2 .ƒ2 /, so after projection on 1;1 ˝ 1;1 it follows that
1
.x y y x/1;1 C .J x Jy Jy J x/1;1 .T .x; y/ C T .J x; Jy//1;1
2
is an element of ƒ2 .1;1 /. We conclude by recording that
.T .x; y/ C T .J x; Jy//1;1 D T 2;2 .x; y/ C T 2;2 .J x; Jy/
since forms in 1;3 are J -anti-invariant whilst those in 2;2 ˚ 4 are J -invariant and
moreover any ' in 4 satisfies '.J ; J ; ; / D '.; ; ; /.
The next step is to notice that (2.5) actually says that R Q belongs to S 2 .ƒ2 /
and to project again on 1;1 ˝ 1;1 . By the argument above it follows that
R.x; y/ C R.J x; Jy/ Ra .x; y/
is in S 2 .1;1 /, thus by Proposition 2.1 we can write
y
y/
R.x; y/ C R.J x; Jy/ Ra .x; y/ D 2RK .x; y/ C 2.x;
(2.6)
for all x, y in V , where is in 2;2 and RK belongs to K.u.m//.
Now, rewriting (2.5) as
R.x; y; z; u/ R.z; u; x; y/ D 2.x .y; z; u/ y .x; z; u// T .x; y; z; u/
11
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
we obtain after projection on 2 ˝ 1;1 that
R.x; y/ R.J x; Jy/ D 2.x y y x/1;1 2.J x Jy Jy J x/1;1
.T .x; y/ T .J x; Jy//1;1
for all x, y in V . Since 4 ˚ 2;2 consists in J -invariant forms, it is easy to see that
.T .x; y/ T .J x; Jy//1;1 D T 1;3 .x; y/ T 1;3 .J x; Jy/;
in other words
1
.R.x; y/ R.J x; Jy// D Rm .x; y/
2
(2.7)
for all x, y in V . The splitting of R follows now from (2.6) and (2.7) and from the
fact that the component of on 1 ˝ 3 is not seen by the projection on 1;1 .
To finish the proof, it remains only to prove the Bianchi identities for Ra and Rm .
Using Lemma 2.3 and (2.2) it is easy to get that
2;2
.b1 Ra /x D 2.x1;2 C J J1;2
x / C 2x ³ T
1
1
x ³ .a. 1;2 / C ac .J 1;2 // J x ³ .ac . 1;2 / a.J 1;2 //
2
2
for all x in V . The Bianchi identity for Rm is proved along the same lines and therefore
left to the reader.
Remark 2.2. Underlying Theorem 2.1 are the following isomorphisms of u.m/modules. The first is
b1 W ƒ2 .1;1 / ! .1 ˝1 1;2 / \ Ker.a/;
where
1 ˝1 1;2 D f 2 1 ˝ 1;2 W J x D J x for all x 2 V g:
The second is given by
b1 W 2 ˝ 1;1 ! 1 ˝2 1;2 ;
where 1 ˝2 1;2 D f 2 1 ˝ 1;2 W J x D J x for all x 2 V g.
This makes that in practice it is not necessary to work with the somewhat involved
expressions for Ra , Rm but rather with their Bianchi contractions, which are tractable.
2.2 The Nijenhuis tensor and Hermitian connections
Let .M 2m ; g; J / be an almost Hermitian manifold. Recall that the Nijenhuis tensor
of the almost complex structure J is defined by
NJ .X; Y / D ŒX; Y ŒJX; J Y C J ŒJX; Y C J ŒX; J Y 12
Paul-Andi Nagy
for all vector fields X, Y on M . It satisfies
NJ .X; Y / C NJ .Y; X / D 0;
NJ .JX; J Y / D NJ .X; Y /;
NJ .JX; Y / D JNJ .X; Y /
for all X, Y in TM . By evaluating NJ using the Riemannian metric g we can also
form the tensor N J defined by
NXJ .Y; Z/ D g.NJ .Y; Z/; X /
for all X, Y , Z in TM , which therefore belongs to 1 ˝2 2 . When NJ vanishes
identically, J is said to be integrable and gives M the structure of a complex manifold.
Denoting by r the Levi-Civita connection attached to the Riemannian metric g
we form the tensor rJ in 1 ˝ 2 . Then one can alternatively compute the Nijenhuis
tensor as
NJ .X; Y / D .rJX J /Y .rJ Y J /X C J .rX J /Y .rY J /X :
(2.8)
It is now a good moment to recall that an almost Hermitian structure .g; J / is called
Kähler if and only if rJ D 0, or equivalently r! D 0. This implies the integrability
of J by making use of (2.8) and also that ! is a symplectic form, in the sense that
d! D 0 where d denotes the exterior derivative. It turns out that for an arbitrary
almost complex structure .g; J /, the tensors d! in ƒ3 and N J in 1 ˝2 2 form a
full set of obstructions to having .g; J / Kähler as the following general fact shows.
Proposition 2.2. For any almost Hermitian structure .g; J / on M we have
J
C X ³ d! C JX ³ Jd!
2rX ! D NJX
(2.9)
for all X in TM .
Proof. Although this is a standard fact we give the proof for self-containedness. From
the definition of the exterior derivative we have
d!.X; Y; Z/ D .rX !/.Y; Z/ .rY !/.X; Z/ C .rZ !/.X; Y /
for all X, Y , Z in TM . Since rX ! is in 2 for all X in TM we obtain
d!.X; Y; Z/ d!.JX; J Y; Z/
D .rX !/.Y; Z/ .rY !/.X; Z/
.rJX !/.J Y; Z/ C .rJ Y !/.JX; Z/ C 2.rZ !/.X; Y /
for all X, Y , Z in TM . This can be rewritten by using (2.8) as
d!.X; Y; Z/ d!.JX; J Y; Z/ D hNJ .X; Y /; J Zi C 2.rZ !/.X; Y /
whenever X, Y , Z belong to TM , and the claim follows.
Therefore d! D 0 and NJ D 0 result in rJ D 0, in other words in .g; J / being
Kähler.
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
13
We shall call a linear connection on the tangent bundle to M Hermitian if it respects both the metric and the almost complex structure. In the framework of almost
Hermitian geometry two connections play a distinguished rôle. The first is called the
first canonical Hermitian connection and it is defined by
xX D rX C X ;
r
where in 1 ˝ 2 is given by X D 12 .rX J /J . The tensor is called the intrinsic
x is
torsion tensor of the U.m/-structure on M induced by .g; J /. The connection r
minimal [28], in the sense of minimising the norm within the space of almost Hermitian
connections.
To introduce the second Hermitian connection we need some preliminaries. First of
all let us decompose the 3-form d! D d 1;2 !Cd 3 ! along the splitting ƒ3 D 1;2 ˚3 .
Then:
Lemma 2.4. Let .M 2m ; g; J / be almost Hermitian. Then a.N J / D 4Jd 3 !.
Proof. Using (2.9) we obtain
d! D a.r!/ D 2m
X
J
ei[ ^ NJe
C 3d! C J.Jd!/;
i
iD1
Since N J belongs to
where fei ; 1 i 2ng is some local orthonormal
P2m [ frame.
J
1
2
J
˝2 it is easy to see that J.a.N // D 3 iD1 ei ^ NJei and the claim follows
by taking into account that J D J on 1;2 .
Since 3 1 ˝2 3 it follows that the Nijenhuis tensor splits as
4
NXJ D NyXJ C X ³ Jd 3 !
3
for all X in TM , where the tensor Ny J belongs to the irreducible U.m/-module
W2 D . ˝2 / \ Ker.a/:
1
2
Proposition 2.3. Let .M 2m ; g; J / be almost Hermitian. The linear connection D
defined by DX D rX C X , where
1
1
2X D X ³ Jd 1;2 ! X ³ Jd 3 ! C NyXJ ;
3
2
is Hermitian.
Proof. That D is metric is clear since X is a two form for all X in TM . To see that
D! D 0 it is enough to show that rX ! C ŒX ; ! D 0 or equivalently
rX ! C JX D 0
ker –> Ker
14
Paul-Andi Nagy
for all X in TM . But
1
1
2JX D J.X ³ Jd 1;2 !/ J.X ³ Jd 3 !/ C J NyXJ
3
2
2
1;2
1;2
J
D X ³ d ! JX ³ Jd ! X ³ d 3 ! C NyJX
3
after taking into account that Ny J belongs to 1 ˝2 2 . Therefore, after taking into
account (2.9),
4
J
2rX ! C 2JX D NyJX
C JX ³ Jd 3 ! C X ³ d! C JX ³ Jd!
3
2
J
X ³ d 1;2 ! JX ³ Jd 1;2 ! X ³ d 3 ! C NyJX
D0
3
after a straightforward computation, and the result follows.
It follows from Proposition 2.2 and Lemma 2.4 that the intrinsic torsion tensor of
the almost Hermitian structure .g; J / is completely determined by
.d!; Ny J / in ƒ3 ˚ W2 :
Since ƒ3 ˚ W2 has four irreducible components under the action of U.m/ the Gray–
Hervella classification [34] singles out 16-classes of almost Hermitian manifolds.
Similar classification results are available for the groups G2 [22], [16] and Spin.7/
[21] as well as for quaternionic structures [41] and SU.3/-structures on 6-dimensional
manifolds [14]. The case of Spin.9/-structures on 16-dimensional manifolds has been
equally treated in [24]. For PSU.3/-structures on 8-dimensional manifolds and SO.3/structures in dimension 5 the decomposition of the intrinsic torsion tensor has been
studied in [37], [58] and [8].
2.3 Admissible totally skew-symmetric torsion
In this section we shall start to specialise our discussion to a particular class of almost
Hermitian manifolds, to be characterized in terms of the torsion tensor of the Hermitian
connection D. We recall that the torsion tensor of a linear connexion on M , e.g. D,
is the tensor T D in ƒ2 ˝ ƒ1 given by
T D .X; Y / D DX Y DY X ŒX; Y x the torsion will be
for all vector fields X, Y on M . In the case of the connection r,
denoted simply by T .
The following result of Friedrich and Ivanov clarifies in which circumstances an almost Hermitian structure admits a Hermitian connection with totally skew-symmetric
torsion.
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
15
Theorem 2.2 ([26]). Let .M 2m ; g; J / be almost Hermitian. There exists a Hermitian
connection with torsion in ƒ3 if and only if N J is a 3-form. If the latter holds, the
connection is unique and equals D.
Almost Hermitian manifolds with totally skew-symmetric Nijenhuis tensor form
the so-called Gray–Hervella class W1 C W3 C W4 and are usually called G1 almost
Hermitian structures. As it follows from the discussion above an almost Hermitian
structure .g; J / belongs to the class G1 if and only if Ny J D 0. Because of its
uniqueness, the connection D in Theorem 2.2 will be referred to as the characteristic
connection of the G1 -manifold .M 2m ; g; J /.
In the remaining part of this section we will work on a given G1 -manifold
.M 2m ; g; J / and we will derive a number of facts to be used later on. We have
4 3
Jd !;
3
1
2 D T D D Jd 1;2 ! Jd 3 !:
3
It is also worth noting that (2.9) is then updated to
NJ D
2
2rX ! D X ³ d 1;2 ! C JX ³ Jd 1;2 ! C X ³ d 3 !
3
for all X in TM . Under a shorter and perhaps more tractable form this reads
C
rX ! D X ³ t C JX ³ J t C X ³
1;2
for all X in TM , where the 3-forms t in C
and
(2.10)
3
in are given by
1
1 1;2
C
d !;
D d 3 !:
2
3
In the same spirit we can also re-express d! and the torsion form as
tD
d! D 2t C 3
C
T D D 2J t C
and
where the 3-form
in 3 is given by
D
C
(2.11)
;
(2.12)
.J ; ; /.
Remark 2.3. (i) When m D 2 any almost Hermitian structure of class G1 is automatically Hermitian, that is NJ D 0. This is due to the vanishing of 3 in dimension 4.
In what follows we shall therefore assume that m 3.
(ii) From (2.10) it is easy to see that almost Hermitian structures in the class G1
are alternatively described as those satisfying .rJX J /JX D .rX J /X for all X
in TM .
When J is integrable it is easy to see that T D D 2Jd! belongs to ƒ3 (actually
after taking into account Lemma 2.4). In this case the connection D is referred
to as the Bismut connection [7].
1;2
16
Paul-Andi Nagy
Almost quaternionic or almost hyperhermitian structures admitting a structure preserving connection with totally skew-symmetric torsion have been introduced and
given various characterisations in [38], [29], [39], [43]. The resulting geometries are
known under the names QKT (quaternion-Kähler with torsion) and HKT (hyperkähler
with torsion).
2.4 Hermitian Killing forms
In this section, briefly recalling the definition of Killing forms in Riemannian geometry
we shall present a variation of that notion, more suitable to the almost Hermitian
setting. This is done to prepare the ground to present, after proving the results in the
next section, the special relationship binding together almost Hermitian structure of
type G1 and this kind of generalised Killing form.
Let .M 2m ; g; J / be almost Hermitian in the class G1 . Using the metric connection
D we can form the associated exterior differential dD . It is given as in the case of the
usual exterior derivative d by
dD D
2m
X
ei[ ^ Dei :
iD1
Given that D is a Hermitian connexion it is easy to see that
ŒJ; dD D .1/p JdD J
(2.13)
?
?
D .1/p JdD
J
ŒJ; dD
(2.14)
holds on ƒp . Its dual reads
?
?
W ƒ? ! ƒ? is the formal adjoint of dD . Note that dD
is computed
on ƒp , where dD
at a point m of M by
2m
X
?
D
ei ³ Dei ;
dD
iD1
where fei ; 1 i 2mg is a local frame around m, geodesic at m w.r.t. the connection D. A straightforward implication of (2.13) is that
dD ' 2 pC1;q ˚ p;qC1
for all ' in p;q . Denoting by 'p;q the orthogonal projection of ' in ƒ? on p;q we
split
dD D @D C @N D ;
where @D ' D .dD '/pC1;q and @N D ' D .dD '/p;qC1 for all ' in p;q .
Lastly, we mention that the Kähler identities
?
ŒdD ; L? D .1/p JdD
J
and
?
ŒdD
; L D .1/pC1 JdD J
(2.15)
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
17
hold whenever ' belongs to ƒp . We can now make the following
Definition 2.1. A form ' in p;q is a Hermitian Killing form if and only if
DX ' D .X ³ A/p;q
for some form A in p;qC1 ˚ pC1;q and for all X in TM .
This is much in analogy with the concept of twistor form from Riemannian geometry, see [53] for details. We shall just recall that that a differential form ' in ƒp .N /,
where .N n ; h/ is some Riemannian manifold, is called a twistor form or conformal
Killing form if
rX ' D
1
1
X ³ d' C
X [ ^ d ?'
pC1
npC1
holds, for all X in TM . Moreover ' is called a Killing form if it also coclosed, that
is d ? ' D 0.
Remark 2.4. Hermitian Killing 1-forms are dual to Killing vector fields for the metric g. This is essentially due to the fact that D has totally skew-symmetric torsion.
In the rest of this section we shall make a number of elementary observations on
Hermitian Killing forms in ? , the most relevant case for our aims.
Proposition 2.4. Let ' in p ; p 2 be a Hermitian Killing form. The following hold:
N
(i) DX ' D .X ³ A/p for all X in TM , where A D @' C 1 @';
pC1
(ii) J' is a Hermitian Killing form.
Proof. (i) From the definition we have
DX ' D .X ³ A/p
for all X in TM , where A is in p;1 ˚ pC1 . We split A D B C C where B and C
belong to p;1 and pC1 respectively. Since for all X in TM we have
1
.JX ³ JB C .p 1/X ³ B/;
2.p 1/
D X ³ C;
.X ³ B/p D
.X ³ C /p
we find, after taking the appropriate contractions that
dD ' D B C .p C 1/C
and the claim follows.
(ii) After applying J to the Hermitian Killing equation satisfied by ', it is enough
to notice that
p
J.X ³ B/p D
.X ³ JB/p
p1
18
Paul-Andi Nagy
holds, together with
J.X ³ C / D
p
X ³ JC
pC1
for all X in TM .
Note that, unlike Riemannian Killing forms, Hermitian Killing ones need not be
necessarily coclosed. This can be easily verified by means of (2.14) and (2.15). The
following identity is useful in order to better understand the notion of Hermitian Killing
form.
Lemma 2.5. Let ' in p ; p 1 be given. Then:
DJX ' DX J ' D 2.X ³ @D J '/p
for all X in TM .
Proof. Let us consider the tensor in 1 ˝1 p given by
X D DJX ' DX J '
for all X in TM . For notational convenience we define B D @D J ' in 1;p and note,
as in the proof of Proposition 2.4 that X 2 TM 7! .X ³ B/p belongs to 1 ˝1 p
as well. Now a straightforward calculation shows that
a.X 2 TM 7! .X ³ B/p / D B:
At the same time one has
a. / D 2@D J '
hence X D 2.X ³ @D J '/p for all X in TM by making use of Lemma 2.1 (iii)
when p ¤ 1. When p D 1 the claim follows by a simple direct verification which is
left to the reader.
Let us now give an equivalent characterisation of Hermitian Killing forms.
Proposition 2.5. The following hold:
(i) a form ' in p ; p 2 is a Hermitian Killing form if and only if the component
of D' on 1 ˝2 p is determined by @N D J ', that is
DJX ' C DX J ' D
2
X ³ @N D J '
pC1
for all X in TM ;
(ii) a form ' in m is a Hermitian Killing form if and only if
j'jD' D
1
1
d j'j2 ˝ ' C Jd j'j2 ˝ J ':
2
2
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
19
Proof. (i) We have
1
1
.DJX ' C DX J '/ .DJX ' DX J '/
2
2
1
D .X ³ @D '/3 C .DJX ' C DX J '/
2
for all X in TM , and the claim follows immediately from (i) in Proposition 2.4.
(ii) If ' is in m , we have that @N D .J '/ D 0 since mC1 D 0. From (i) we know
that ' is a Hermitian Killing form if and only if
DX J ' D
DJX ' C DX J ' D 0
(2.16)
holds for any X in TM . Let U be the open subset of M given by U D fm 2 M W
'm ¤ 0g and let Z be the complement of U in M . We claim that (2.16) holds iff it
holds on U . Indeed (2.16) holds trivially on int.Z/ and U [ int.Z/ is dense in M .
Suppose now that ' in m is a Hermitian Killing form. Then ', J ' is a basis of
m
jU , hence
D' D a ˝ ' C b ˝ J '
for a couple of 1-forms a, b on U . The first is determined by j'ja D 12 d j'j2 . On the
other hand the fact that ' is a Hermitian form implies that b D Ja, hence our claim
is proved on U and by the density argument above on M . The converse statement is
also clear from the previous observations.
An immediate consequence of (ii) in the proposition above is that a Hermitian
Killing form in m is parallel with respect to the connection D as soon as it has
constant length. It is now a good moment to provide some examples of Hermitian
Killing forms.
Proposition 2.6. Let .M 2m ; g; J / be a Kähler manifold and let k , k D 1; 2 be
holomorphic Killing vector fields, that is
Lk g D 0 and Lk J D 0 for k D 1; 2:
The form ' D .1[ ^ 2[ /2 is a Hermitian Killing form.
Proof. The Killing equation yields
1
X ³ d k[
2
for all X in TM and k D 1; 2. After a few manipulations we get
rX k[ D
1
1
.X ³ d 1[ / ^ 2[ C 1[ ^ .X ³ d 2[ /
2
2
1
1
1
D X ³ d.1[ ^ 2[ / h2 ; X id 1[ C h1 ; X id 2[
2
2
2
rX .1[ ^ 2[ / D
20
Paul-Andi Nagy
for all X in TM . Since our given vector fields are also holomorphic, the forms d k[ ,
k D 1; 2 belong to 1;1 and the claim is proved by projecting onto 2 while using the
N D 0 because one has, as is well known, d.J [ / D 0
Definition 2.1. We note that @'
k
for k D 1; 2.
We continue by giving an exterior algebra characterisation of Hermitian Killing
forms, in the context of a G1 -manifold .M 2m ; g; J /.
Proposition 2.7. Let p 3 be odd. Then:
(i) if ' in p is a Hermitian Killing form, then
2
' • @N D J 'I
pC1
?
?
?
dD
.' ^ J '/ D .dD
'/ ^ J ' ' ^ dD
J' (ii) if ' in p satisfies
?
?
?
dD
.' ^ J '/ D .dD
'/ ^ J ' ' ^ dD
J' 2
' • @N D J '
pC1
and it is everywhere nondegenerate, then ' is a Hermitian Killing form.
Proof. We have
ei ³ Dei .' ^ J '/ D ei ³ .Dei ' ^ J ' C ' ^ Dei J '/
D .ei ³ Dei '/ ^ J ' Dei ' ^ .ei ³ J '/
C .ei ³ '/ ^ Dei J ' ' ^ .ei ³ Dei J '/:
After summation, we get
?
?
?
.' ^ J '/ D .dD
'/ ^ J ' C ' ^ dD
J'
dD
2m
X
Dei ' ^ .ei ³ J '/ C
kD1
2m
X
.ei ³ '/ ^ Dei J ':
kD1
But
2m
X
kD1
Dei ' ^ .ei ³ J '/ D
2m
X
Dei ' ^ .Jei ³ '/
kD1
D
2m
X
DJei ' ^ .ei ³ '/ D kD1
2m
X
.ei ³ '/ ^ DJei '
kD1
whence
?
?
?
.' ^ J '/ D .dD
'/ ^ J ' C ' ^ dD
J'
dD
C
2m
X
kD1
ei ³ ' ^ .DJei ' C Dei J '/:
(2.17)
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
21
(i) If ' is a Hermitian Killing form one uses (2.17) and (i) in Proposition 2.5 to obtain
the conclusion.
(ii) In this case, using (2.17) again we have that
2m
X
.ei ³ '/ ^ ei D 0;
iD1
where in ˝2 is given by
1
p
X D DJX ' C DX J ' 2
X ³ @N D J '
pC1
for all X in TM . We define now O in p1 ˝ p by
.X
O 1 ; : : : ; Xp1 / D X1 ³³Xp1 ³ '
whenever Xk , 1 k p1 belong to TM . It is easy to verify that O is in p1 ˝1 p ,
and since a./
O D 0 Lemma 2.1 (iii) yields O D 0. Because ' is everywhere nondegenerate we obtain that vanishes and we conclude by Proposition 2.5 (i).
To end this section let us present another way of characterizing Hermitian Killing
forms, this time under the form of a product rule with respect to the exterior differential
dD .
Proposition 2.8. Let p 3 be odd. We have:
(i) any Hermitian Killing form ' in p satisfies
dD .' • '/ D dD ' • ' C dD J ' • J ' 2
J.' • @N D J '/I
pC1
(ii) if ' in p satisfies
dD .' • '/ D dD ' • ' C dD J ' • J ' 2
J.' • @N D J '/
pC1
and it is nowhere degenerate, then ' is a Hermitian Killing form.
Proof. We will mainly use Proposition 2.7 and the Kähler identities. Indeed, let us
observe that
L? .' ^ J '/ D ' • ':
Using (2.15) it follows that
?
dD L? .' ^ J '/ D L? dD .' ^ J '/ C .JdD
J /.' ^ J '/
hence
?
.' ^ J '/:
dD .' • '/ D L? dD .' ^ J '/ C JdD
It is easy to see that
L? .dD ' ^ J '/ D .L? dD '/ ^ J ' C dD ' • ';
L? .' ^ dD J '/ D .L? dD J '/ ^ ' C dD J ' • J ';
22
Paul-Andi Nagy
hence we get further
dD .' • '/ D .L? dD '/ ^ J ' C dD ' • '
?
.' ^ J '/:
.L? dD J '/ ^ ' C dD J ' • J ' C JdD
Suppose now that ' is a Hermitian Killing form. Then, by using (i) in Proposition 2.7
we get
?
?
?
dD
.' ^ J '/ D dD
' ^ J ' ' ^ dD
.J '/ 2
' • @N D .J '/
pC1
while the second part of the Kähler identities (2.15) provides us with
?
L? dD .J '/ D dD
';
?
?
L dD ' D dD J ':
The claim in (i) follows now by a simple computation. The converse statement in (ii)
is proved by using methods similar to those employed for (ii) in Proposition 2.7, and
it is therefore left to the reader.
3 The torsion within the G1 class
3.1 G1 -structures
In the following .M 2m ; g; J / will be almost-Hermitian, in the class G1 . The connection D acts on any form ' in ƒ? according to
1
DX ' D rX ' C ŒX ³ T D ; '
2
for all X in TM . Based on this fact it is straightforward to check that the differential
dD is related to d by
dD ' D d' 2m
X
.ek ³ T D / ^ .ek ³ '/
kD1
?
for all ' in ƒ . This yields, in the particular case of 3-forms the following comparison
formula
dD ' D d' T D • '
(3.1)
for all ' in ƒ3 .
Let now R be the curvature tensor of the metric g with the convention that
2
2
C rY;X
for all X , Y in TM . In the same time we consider
R.X; Y / D rX;Y
D
the curvature tensor R of the connection D where the same convention applies. We
shall now recall that the first Bianchi identity for the connection D takes the following
form.
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
23
Lemma 3.1. For all X in TM ,
1
b1 .RD /X D DX T D C X ³ d T D :
2
Proof. It is easy to see, directly from the definition, that the curvature tensors of the
Levi-Civita connection and of the connection D are related by
1 D
1
RD .X; Y / D R.X; Y / Y ³ DX T D X ³ DY T D C "T .X; Y / (3.2)
2
4
for all X, Y in TM . Here we have set
D
"T .X; Y / D ŒTXD ; TYD 2TTDD Y :
X
Since R satisfies the algebraic Bianchi identity and
D
b1 ."T /X D 2X ³ .T D • T D /;
the assertion follows eventually by setting X D ek , taking the wedge product with ek
in (3.2) and summing over 1 k 2m. Note that in the process one also uses the
comparison formula (3.1).
Let us gather now some information on the differentials of the components of the
torsion form T D .
Proposition 3.1. Let .M 2m ; g; J / belong to the class G1 . The following hold:
(i) @D t D 0;
(ii) 3@D
(iii) @N D
C
C
C 2@N D t D 4.t • J t / 4.J t •
D 83 .J t •
C
C
/1;3 ;
/4 :
C
Proof. Since d! D 2t C 3
is closed it follows that
2dt C 3d
C
D 0:
Rewritten by means of the connection D this identity yields, when also using the
comparison formula (3.1),
2.dD t C T D • t / C 3.dD
C
We now take into account that T D D 2J t C
product and use of Lemma 2.2 (v) at
2dD t C 4J t • t C 2
•
t
An elementary computation yields J.
.
.
•
t /4
•
t /1;3
C 3dD
•
t/
C
CTD •
to arrive, after expansion of the
C
C 6J t •
D Jt •
D .J t •
D .J t •
/ D 0:
C
C
D 0:
, in particular
/4 ;
C
C
/1;3 :
24
Paul-Andi Nagy
It suffices thus to use Lemma 2.2 to obtain the proof of the claims after identifying the
various components in the equation above along the bidegree decomposition of ƒ4 .
We are now ready to prove the main result of this section.
Theorem 3.1. Let .M 2m ; g; J / be almost Hermitian, of type G1 . Then
mitian Killing form, that is
DX
D X ³ .@D
1
C @N D
4
is a Her-
/
3
for all X in TM .
Proof. Since D is a Hermitian connection we have ŒRD .X; Y /; ! D 0, in other words
2m
X
RD .X; Y /ei ^ .Jei /[ D 0
iD1
for all X, Y in TM . Setting X D ek and taking the exterior product with ek we find
2m
X
ek[ ^ RD .ek ; Y /ei ^ .Jei /[ D 0:
i;kD1
Obviously, this is equivalent with
2m
X
ei ³ .ek[ ^ RD .ek ; Y // ^ .Jei /[ D
2m
X
RD .ei ; Y / ^ .Jei /[
iD1
i;kD1
or further, by using Lemma 3.1,
X
1
C X ³ dD T D / D
RD .ei ; Y / ^ .Jei /[
2
2m
J.DX T
D
iD1
for all Y in TM . Since D is Hermitian, we have that RD .X; Y / is in 1;1 , hence the
right-hand side above is in 1;2 given that ƒ1 ^ 1;1 1;2 . Thus
DX T
D
1
C X ³ dT D
2
3
D 0;
and it follows that
1
(3.3)
D .X ³ d T D /3
2
is a Hermitian Killing form and the claim follows by
DX
for all X in TM . Therefore,
using Proposition 2.4.
We have the following immediate consequence of the above result.
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
25
Theorem 3.2. Let .M 2m ; g; J / have totally skew-symmetric Nijenhuis tensor. Then
DNJ D 0 if and only if d T D belongs to 2;2 :
In particular NJ is parallel if T D is a closed 3-form.
Proof. This is a direct consequence of (3.3) and of the properties of the projection
on 3 .
Remark 3.1. It also follows from (3.3) when combined with Theorem 3.1 that d T D
belongs to 2;2 if and only if dD C D 0.
In dimension 6, the fact that C is a Hermitian Killing form is described by (ii)
in Proposition 2.5. As observed in [13], it amounts then to the local parallelism of
C
w.r.t. the characteristic connection, after performing a conformal transformation in
order to normalise the length of C to a constant. In dimension 6 structures of type G1
with DT D D 0 have been classified in [4]. Classification results are also available [2]
under the same assumption in the Hermitian case, provided that the holonomy of D
is contained in S 1 U.m 1/.
3.2 The class W1 C W4
W1 C W4 structure In this section we shall record some of the additional features of
the geometry of almost Hermitian manifolds in the Gray–Hervella class W1 C W4 . It
can be described as the subclass of G1 having the property that t has no component on
ƒ30 , or alternatively
t D ^ !:
(3.4)
Here is a 1-form on M , called the Lee form of the almost-Hermitian structure .g; J /.
It can be recovered directly from the Kähler form of .g; J / by
.m 1/J D d ? !;
in particular we have that d ? .J / D 0.
Proposition 3.2. Let .M 2m ; g; J /; m 3 be almost-Hermitian in the class W1 C W4 .
The following hold:
(i) @D D 0;
(ii) @D
C
D 2.J ^
1N
@
4 D
C
D^
(iii)
C
C
C ^
J ^
/ C 23 .@N D 2 ³
/ ^ !;
.
Proof. All claims follows from Proposition 3.1, applied to our present situation, that
is t D ^ !. Therefore, (i) is immediate from (i) in the previously cited proposition.
26
Paul-Andi Nagy
Through direct computation we find
.J ^ !/ • C D . ³
.J ^ !/ •. ^ !/ D 0:
/ ^ ! C 3J ^
Our last two assertions follow now from Proposition (3.1), after projection of the
formulae above onto ƒ4 D 2;2 ˚ 1;3 ˚ 4 .
In dimension 6 it has been shown in [13], [16] that an almost Hermitian manifold
in the class W1 C W4 has closed Lee form, i.e., d D 0. Theorem 3.1 combined with
Proposition 3.2 seems a good starting point to investigate up to what extent structures of
W1 C W4 are closed in arbitrary dimensions but we shall not pursue this direction here.
3.3 The u.m/-decomposition of curvature
In this section we shall investigate, for further use, the splitting of the curvature tensor
of the Hermitian connection D. Our main goal is to identify explicitly the "non-Kähler"
part of the tensor RD and to give it an explicit expression in terms of the torsion form
T D of the characteristic connection. Note that for almost quaternionic-Hermitian or
G2 -structures (not necessarily with skew-symmetric torsion) similar results have been
obtained in [42], [15]. We will mainly use that
RD .X; Y; Z; U / RD .Z; U; X; Y /
1
D .DX T D /Y .Z; U / .DY T D /X .Z; U /
(3.5)
2
1
C .DZ T D /U .X; Y / .DU T D /Z .X; Y /
2
holds for all X; Y; Z; U in TM , as it easily follows from the difference formula (3.2),
D
after verifying that "T belongs to S 2 .ƒ2 /.
Theorem 3.3. Let .M 2m ; g; J / be almost Hermitian of type G1 . We have
y C 1 Ra C Rm ;
RD D RK C 2
where RK belongs to K.u.m//. Moreover:
(i) Ra belongs to ƒ2 .1;1 / and satisfies the Bianchi identity
1
1
.b1 Ra /X D DX .J t / DJX t X ³ @D .J t /
2
2
for all X in TM .
(ii) The Bianchi identity for Rm in 2 ˝ 1;1 is
1
.b1 Rm /X D DX .J t / C DJX t C .JX ³ J@D
4
for all X in TM .
2X ³ @D
/
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
27
(iii) in 2;2 is given by
D
3
1
@D .J t / C 2.J t • J t /2;2 C
2
2
•
:
Proof. This is a direct application of Theorem 2.1 which can be used, as it follows
from (3.5), for the tensor R D RD and
1
1
D DT D D D.J t / D
2
2
:
Then we have 1;2 D D.J t / and 3 D 12 D , and in order to prove our claims
we only need to determine the various antisymmetrisations of 1;2 . Also note that in
this context
1
1
T D dD T D D dD .J t / dD ;
2
2
in particular T 2;2 D @D .J t/.
(i) Now, it is easy to see that
a. 1;2 / D dD .J t /;
a.J 1;2 / D dD t;
ac . 1;2 / D JdD t;
ac .J 1;2 / D JdD J t:
Hence,
A1 D dD .J t/ JdD .J t / 4T 2;2 D dD .J t / JdD .J t / C 4@D .J t /
D 2@D .J t/ C 4@D .J t / D 2@D .J t /
and
A2 D JdD t dD t D 2@D t D 0
by (i) of Proposition 3.1.
(ii) Since T 1;3 D @N D .J t / 12 @D
we have
A3 D dD .J t/ C JdD J t C 2@N D .J t / C @D
D @D
1
1
A4 D dD t JdD t C J @N D .J t / C J@D D J@D
2
2
after making use of (2.13). The claim in (ii) now follows.
(iii) We apply the Bianchi operator to
and
y C Ra C Rm
RD D RK C and find after making use of (i), (ii) and of (2.2) that
1
1
b1 .RD /X D X ³ C 2DX .J t / X ³ @D .J t / C .JX ³ J@D
2
4
2X ³ @D
/
28
Paul-Andi Nagy
for all X in TM . Plugging into this the Bianchi identity for D in Lemma 3.1, combined
with the fact that is a Hermitian Killing form as asserted in Theorem 3.1 yields
further
1
1
1
.X ³ d T D /1;2 D X ³ @D .J t / C .JX ³ J@D 2X ³ @D /
2
2
4
1
D X ³ @D .J t / .X ³ @D C /1;2
2
for all X in TM . After identifying components on 1 ˝1 1;2 and 1 ˝2 1;2 respectively (see also Remark 2.1 for the definition of these spaces) we find that
1
1
.d T D /2;2 D @D .J t /:
2
2
Now d T D D dD T D C T D • T D by the comparison formula (3.1) thus by projection
on 2;2 we find
.d T D /2;2 D 2@D .J t / C .T D • T D /2;2
D 2@D .J t / C 4.J t • J t /2;2 C
•
after expansion of the product and use of Lemma 2.2 (iii). The claim in (iii) follows
now immediately.
Remark 3.2. The curvature decomposition in Theorem 3.3 can be still refined, given
that the U.m/-modules
K.u.m//;
2;2 ;
ƒ2 .1;1 /
and
2 ˝ 1;1
are not irreducible. Although this is an algebraically simple procedure, the computations at the level of the derivative DT D of the torsion form become somewhat involved
and will not be presented here. We just illustrate the situation in the simpler case of
W1 C W4 below.
Theorem 3.3 has various applications, a class of which consists in giving torsion
interpretation of curvature conditions imposed on the tensors RD or R. As an example
in this direction we have the following:
Proposition 3.3. Let .M 2m ; g; J / belong to the class G1 . The curvature tensor RD
is Hermitian, that is
RD .JX; J Y / D RD .X; Y /
for all X, Y in TM , if and only if
DJX t C DX .J t / D
for all X in TM and
2
.X ³ @N D .J t //1;2
3
2@N D .J t / D 3@D
:
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
29
Proof. By Theorem 3.3 the tensor RD is Hermitian if and only if Rm D 0 which in
turn happens if and only if b1 Rm D 0. Therefore, by (ii) in Theorem 3.3 the curvature
of D is Hermitian if and only if
1
DX .J t/ C DJX t C .JX ³ J@D 2X ³ @D
4
for all X in TM . Taking the alternation above we find
2@N D .J t / 3@D
/D0
D 0:
We conclude by recalling that
.X ³ A/1;2 D
1
.2X ³ A JX ³ JA/
4
(3.6)
for all X in TM , where A belongs to 1;3 .
Remark 3.3. In the case when the curvature tensor of the connection D is Hermitian
the fact that 2@N D .J t/ D 3@D combined with (ii) in Proposition 3.1 further yields
@N D .J t / D .t • J t / .J t •
C
/1;3 :
For the subclass W1 ˚ W4 G1 more information on the curvature tensor of the
connection D is available and it will actually turn out that the components Ra and Rm
have simple algebraic expressions. We define
S 2; .M / D fS 2 S 2 .M / W SJ C JS D 0g:
This is embedded in 2 ˝ 1;1 via S 7! SV where
1
V
S.X;
Y / D ..SJX/[ ^ Y [ C X [ ^ .SJ Y /[ C .SX /[ ^ .J Y /[ C .JX /[ ^ .S Y /[ /
2
for all X, Y in TM . One verifies that
.b1 SV /X D .SX /[ ^ !
(3.7)
z
for all X in TM . We also have an embedding ,! ˝ given by 7! where we define
1
z
.X;
Y / D ..JX; J Y / .X; Y //
4
for all X, Y in TM . Elementary considerations ensure that this is well defined and
subject to
z X D .X ³ /1;2
(3.8)
.b1 /
1;3
2
1;1
whenever X belongs to TM . As a last piece of notation let the symmetrized action of
D on 1-forms be defined by
1
V
.D˛/.X;
Y / D ..DX ˛/Y C .DY ˛/X /
2
whenever ˛ is in 1 and X, Y belong to TM .
30
Paul-Andi Nagy
Proposition 3.4. Let .M 2m ; g; J / belong to the class W1 C W4 . If denotes the Lee
form of .g; J / the following hold:
(i) D 32 @D .J / C 2jj2 ! 4 ^ J ^ ! C 12 • ;
(ii) Ra D @D .J / ˝ ! ! ˝ @D .J /;
A , where S
(iii) Rm D @N D .J / ˝ ! C SV @D
V /.
S D .1 J /D.J
in S 2; .M / is defined by
Proof. Recall that for the class W1 C W4 we have that t D ^ !. Taking this into
account we will apply now Theorem 3.3.
(i) follows from (iii) in Theorem 3.3 when observing that
.J t • J t /2;2 D jj2 ! ^ ! 2! ^ ^ J :
(ii) First of all we note that
DX .J / DJX D X ³ @D .J /
for all X in TM . The quickest way to see this is to observe that the tensor q in
1 ˝ 1 defined by q.X/ D DX .J / DJX X ³ @D .J / for all X in TM is
actually in 1 ˝1 1 and satisfies a.q/ D ac .q/ D 0, by using also that @D D 0
(cf. Proposition 3.2 (i)). Our claim now follows by observing that (iii) in Lemma 2.1
continues to hold when p D q D 1.
Using now (i) in Theorem 3.3 we get
.b1 Ra /X D .X ³ @D .J // ^ ! @D .J / ^ .X ³ !/
for all X in TM .
Now for any ' in 1;1 we consider the element ! ˝ ' ' ˝ ! in ƒ2 .1;1 /, which
is explicitly given by
.! ˝ ' ' ˝ !/.X; Y / D !.X; Y /' '.X; Y /!
for all X , Y in TM . A straightforward computation following the definitions yields
b1 .@D .J / ˝ ! ! ˝ @D .J //X D .X ³ @D .J // ^ ! @D .J / ^ .X ³ !/
for all X in TM , and we conclude by using the injectivity of the Bianchi map
b1 W ƒ2 .ƒ2 / ! ƒ1 ˝ ƒ3 .
(iii) By (iii) in Theorem 3.3 we have
.b1 Rm /X D DX .J / C DJX ^ ! .X ³ @D /1;2
after also making use of (3.6). Since
DX .J / C DJX D X ³ @N D .J / C .S X /[
A belongs to
it follows by means of (3.7), (3.8), that Rm C @N D .J / ˝ ! SV C @D
Ker.b1 / \ .2 ˝ 1;1 /. It therefore vanishes and the proof is finished.
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
31
Finally, we remark that Rm can be given a more detailed expression by taking into
account (ii) of Proposition 3.2.
4 Nearly-Kähler geometry
We shall begin here our survey of nearly-Kähler geometry. This is the class of almost
Hermitian structures introduced by the following.
Definition 4.1. Let .M 2m ; g; J / be an almost Hermitian manifold. It is called nearlyKähler (NK for short) if and only if
.rX J /X D 0
whenever X belongs to TM .
This can be easily rephrased to say that an almost Hermitian structure .g; J / is
nearly-Kähler if and only if its Kähler form is subject to
1
X ³ d!
(4.1)
3
for all X in TM . In other words, the Kähler form of any NK-structure is a Killing
form and conversely U. Semmelmann [53] shows that any almost Hermitian structure
with this property must be nearly-Kähler. Therefore, nearly-Kähler structures belong
to the class W1 in the sense that the 3-form t in Proposition 2.2 vanishes identically.
From now on we shall work on a given nearly-Kähler manifold .M 2m ; g; J /. For
many of the properties of an NK-structure are best expressed by means of its first
canonical Hermitian connection, it is important to note then the coincidence of the
canonical connection and D, that is
rX ! D
x
D D r:
For this reason, the torsion tensor T D will be denoted simply by T in what follows.
It is given by
TD D and hence belongs to 3 . Therefore, in dimension 4 NK-structures are Kähler and we
shall assume from now on that m 3. The Nijenhuis tensor of the almost complex J
is computed from (2.8) by
N J D 4 :
As an immediate consequence one infers that the almost complex structure of an
NK-manifold is integrable if and only if the structure is actually a Kähler one. This
observation motivates the following
Definition 4.2. Let .M 2m ; g; J / be an NK-structure. It is called strict iff rX J D 0
implies that X D 0 for all X in TM .
32
Paul-Andi Nagy
Therefore the non-degeneracy of any of the forms C and is equivalent with
the strictness of the NK-structure .g; J /. An important property of NK-structures is
contained in the following result.
Theorem 4.1. Let .M 2m ; g; J / be a nearly-Kähler manifold. Then
x
r
˙
D0
in other words the Nijenhuis tensor of J is parallel w.r.t. the canonical Hermitian
connection.
This has been proved first by Kirichenko [40] and a short proof can be found in
[5]. It also follows from our Theorem 3.1, which is unifying this type of property in
the class G1 .
Let us introduce now the symmetric tensor r in S 2 M given by
hrX; Y i D hX ³
C
;Y ³
C
i
for all X, Y in TM . It is easily seen to be J -invariant and if .M 2m ; g; J / is strict
then r is non-degenerate. Moreover, from Theorem 4.1 it also follows that
x D 0:
rr
Corollary 4.1 ([48]). Any NK-manifold is locally the product of a Kähler manifold
and a strict nearly-Kähler one.
x
Proof. Consider the r-parallel
distribution V WD fV 2 TM W VC D 0g. Since the
x vanishes in direction of V the latter must be parallel for
torsion of the connection r
the Levi-Civita and the result follows by using the de Rham splitting theorem.
It follows that locally and also globally if our original manifold is simply connected
we can restrict to the study of strict nearly-Kähler structures (SNK for short). Note
however that in dimension 6, any NK-structure which is not Kähler must be strict.
x is eventually reflected in the properties
The parallelism of the torsion tensor w.r.t. r
its curvature tensor. Indeed
Proposition 4.1 ([30], [33]). The following hold:
x
x
(i) R.X;
Y; Z; U / D R.Z;
U; X; Y / for all X; Y; Z; U in TM ;
x
x
(ii) R.JX; J Y / D R.X; Y / for all X , Y in TM ;
x
x Z/X C R.Z;
x
(iii) R.X;
Y /Z C R.Y;
X/Y D Œ
Z in TM .
X;
Y Z
XY
Z for all X , Y ,
Proof. (i) is immediate from (3.5) and the parallelism of the torsion, whilst (ii) follows
x belongs to ƒ2 ˝ 1;1 . The last claim follows for instance
from (i) and the fact that R
from Lemma 3.1
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
33
Remark 4.1. Proposition 4.1 continues to hold when the nearly-Kähler metric is
allowed to have signature. This has been exploited in [18] to classify NK-structures
compatible with a flat pseudo-Riemannian metric.
To establish first order properties of SNK-structures we will have a look at the
Ricci tensor of such metrics. The Hermitian Ricci tensor of .g; J / is defined by
hRicX; Y i D
2m
X
x
R.X;
ei ; Y; ei /
iD1
for all X, Y in TM , where fei ; 1 i 2mg is some local orthonormal frame. By
making use of Proposition 4.1 we find that Ric is actually symmetric and J -invariant.
The Hermitian Ricci tensor is related to the usual Riemannian one by
3
Ric D Ric r
4
as implied by the general curvature comparison formula (3.2).
(4.2)
Theorem 4.2 ([48]). Let .M 2m ; g; J / be a strict nearly-Kähler manifold. The following hold:
x that is rRic
x
(i) the Ricci tensor of the metric g is parallel w.r.t. r,
D 0;
(ii) Ric is positive definite.
Proof. The proof of both relies on the explicit computation of the Ricci tensor of the
metric g. From the parallelism of C , after derivation and use of the Ricci identity
x we find
for the connection with torsion r
x
ŒR.X;
Y /;
C
D0
(4.3)
for all X, Y in TM . In a local orthonormal frame fei ; 1 i 2mg this reads
2m
X
x
R.X;
Y /ei ^
C
ei
D0
iD1
for all X, Y in TM . We now set Y D ek , take the interior product with ek above to
find, after summation over 1 k 2m and some straightforward manipulations that
X
C
x
D
R.X;
ek /ei ^ eCi ;ek :
RicX
1k;i2m
C
is a form, the sum in the right-hand side equals
Since
1 X
x
x
.R.X;
ek /ei R.X;
ei /ek / ^ eCi ;ek
2
1k;i2m
D
1
2
X Œ
1k;i2m
C
X;
C
ek ei
C
C
X ek
ei ^
C
ei ;ek
C
1
2
X
x i ; ek /X ^
R.e
1k;i2m
C
ei ;ek
34
Paul-Andi Nagy
x Now the last sum vanishes since C is J -antiafter using the Bianchi identity for R.
x is J -invariant and our frame can be chosen
invariant in the first arguments whereas R
to be Hermitian. After neglecting terms which are J -invariant in ei ; ek in the algebraic
sum above we end up with
1 X
C
C C
C
D
X ek ei ^ ei ;ek
RicX
2
1k;i2m
for all X in TM . Using now the definition of the tensor r a straightforward manipulation yields
2m
1X C
C
[
D
(4.4)
X ek ^ .rek /
RicX
2
kD1
for all X in TM . By derivation and using the parallelism of
C
xX Ric/Y
.r
C
it follows that
D0
for all X, Y in TM . (i) follows now from the fact that C is nondegenerate and the
comparison fact in (4.2). For the claim in (ii) we refer the reader to [48].
Using Myer’s theorem it follows from the above that complete SNK-manifolds
must be compact with finite fundamental group. Therefore, in the compact case one
can restrict attention, up to a finite cover, to simply connected nearly-Kähler manifolds.
Another important object is the first Chern form of the almost Hermitian structure
.g; J / defined by
2m
X
x
R.X;
Y; ei ; Jei /
81 .X; Y / D
iD1
x is J -invariant we have that 1 belongs to 1;1 so one can
for all X, Y in TM . Since R
write 41 D hCJ ; i for some C in S 2 .TM / such that CJ D J C . A straightforward
computation using the first Bianchi identity yields the relation
C D Ric r;
hence C must be parallel w.r.t. the canonical connection, that is
x D0
rC
by means of Theorem 4.2. Since 1 is a closed form, as it follows from the second
x this results in the algebraic obstruction
Bianchi identity for r,
C.
C
XY/
D
C
X CY
C
C
CX Y
(4.5)
for all X, Y in TM . Note that this can be given a direct algebraic proof by observing
x belongs
that Œ1 ; C D 0, as it follows from (4.3) when taking into account that R
2
2
to S .ƒ /.
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
35
4.1 The irreducible case
To obtain classification results for strict nearly-Kähler manifolds we shall start from
examining the holonomy representation of the canonical Hermitian connection. At a
point x of M where .M 2m ; g; J / is some SNK-manifold this is the representation
x W Tx M ! Tx M
Holx .r/
x along loops about x. The holonomy representaobtained by parallel transport w.r.t. r
x is a Hermitian connection and this gives rise to two different
tion is Hermitian, for r
notions of irreducibility as indicated by the following well-known result from representation theory.
Proposition 4.2. Let .V 2m ; g; J / be a Hermitian vector space and let .G; V / be a
Hermitian representation of some group G. If we write V C for the complex vector
space obtain from V by setting iv D J v for all v in V , the following cases can occur:
(i) .G; V / is irreducible.
(ii) .G; V C / is irreducible but not .G; V /. In this case V splits orthogonally as
V D L ˚ JL for some G-invariant subspace L of V .
(iii) .G; V C / is reducible.
In this section we shall deal with the instances when the holonomy representation
x
of r is irreducible in the sense of (i) or (ii) in the Proposition 4.2. The first is actually
covered by the following powerful result of R. Cleyton and A. Swann.
Theorem 4.3 ([17]). Let .N n ; g/ be Riemannian such that there exists a metric connection D such that the following hold:
(i) the torsion tensor T of D belongs to ƒ3 ;
(ii) DT D 0 and T does not vanish identically.
If the holonomy representation of D is irreducible then D is an Ambrose–Singer
connection in the sense that DRD D 0 where RD denotes the curvature tensor of the
connection D, provided that n ¤ 6; 7.
The two exceptions in the result above correspond actually to nearly-parallel G2 structures in dimensions 7 (see [27] for an account) and SNK-structures in dimension 6.
In the situation in the theorem above .N n ; g/ is a locally homogeneous space (see
[57]) for more details). Theorem 4.3 is proved by making use of general structure
results on Berger algebras and formal curvature tensors spaces, for irreducible representations of compact Lie algebras (see also [44] for the non-compact case). We can
now state the following.
Theorem 4.4. Let .M 2m ; g; J / be a strict nearly-Kähler manifold. Then either:
x is an Ambrose–Singer connection; or
(i) r
36
Paul-Andi Nagy
(ii) m D 3; or
x is reducible over C.
(iii) the holonomy representation of r
Proof. By making use of Theorem 4.3 we see that the case (i) in Proposition 4.2
corresponds to (i) in our statement. To finish the proof let us suppose that at some
point x of M we have an orthogonal splitting
Tx M D Lx ˚ JLx
x
for some Holx .r/-invariant
subspace of Tx M . Using parallel transport Lx extends
x
to a r-parallel
distribution L of TM such that
TM D L ˚ JL:
x
x belongs to
It follows that R.L;
JL/ D 0 by also using Proposition 4.1 (ii). Since R
2 1;1
C
x
, which can be explicitly
S . /, this means that R is an algebraic expression in
determined from the first Bianchi identity (see [49] for details). It follows that that
xR
x is an Ambrose–Singer connection.
x D 0 and then r
r
5 When the holonomy is reducible
In this section we present classification results for SNK-structures in the case when
x is complex reducible. We begin by setting up
the holonomy representation of r
some terminology which is aimed to gain some understanding concerning the relation
between the algebraic properties of the torsion form of an SNK-structure and the
geometry of the underlying Riemannian manifold.
5.1 Nearly-Kähler holonomy systems
We start by the following definition which extracts the more peculiar facts from NKgeometry which relate to the holonomy of the canonical Hermitian connection.
Definition 5.1. A strict nearly-Kähler holonomy system .V 2m ; g; J;
stituted of the following data:
C
; R/ is con-
(i) a Hermitian vector space .V 2m ; g; J /;
(ii) a nondegenerate 3-form
C
in 3 ;
(iii) a tensor R in 1;1 ˝ 1;1 of the form
y
R D RK C ;
where RK in K.u.m// is such that
ŒR.x; y/;
C
D0
(5.1)
37
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
holds for any x, y in V . Moreover the form in 2;2 must be given by D
1 C• C
.
2
Geometrically, the assumption of having the form C nondegenerate amounts to
working on an SNK-manifold (see also Definition 4.2 and Corollary 4.1). In what
follows we shall work on a given strict nearly-Kähler holonomy system to be denoted
by .V 2m ; g; J; C ; R/. As with Riemannian holonomy systems, the liaison with the
holonomy algebra of the canonical connection of a geometric SNK-structure is through
the Lie subalgebra
h WD LiefR.x; y/ W x; y 2 V g
of 1;1 .
Definition 5.2. An SNK-holonomy system .V 2m ; g; J;
if the representation .h; V C / is reducible.
C
; R/ is complex reducible
Another object of relevance here is the isotropy algebra g of the form
by
g D f˛ 2 ƒ2 W Œ˛;
C
C
defined
D 0g:
It is easy to see, starting from (2.1) and then using an invariance argument together
with the non-degeneracy of C that
g 1;1 :
Moreover, the condition (5.1) in Definition 5.1 reads h g.
We are interested here in the structure of complex invariant subspaces of the metric
representation .h; V /. The following definition singles out three main classes of
subspaces of relevance for our situation.
Definition 5.3. A proper, J -invariant subspace V of V is said to be (w.r.t.
(i) isotropic if C .V ; V / V;
(ii) null if
C
C
):
.V ; V / D 0;
(iii) special if it is null and
of V in V .
C
.H; H / D V, where H is the orthogonal complement
Remark 5.1. Any 2-dimensional, J -invariant subspace of V is null w.r.t. C . Moreover, in dimension 6, any two dimensional J -invariant subspace is special w.r.t. C .
However, we are interested here in isotropic or special subspaces which are invariant
w.r.t. a particular Lie group or Lie algebra.
A useful criterion to prove that a subspace is special is the following.
Lemma 5.1. Let .V 2m ; g; J / be a Hermitian vector space and let C be nondegenerate in 3 . If V V is null w.r.t. C and such that C .H; H / V,
then:
38
Paul-Andi Nagy
C
(i) V is special w.r.t.
(ii)
C
;
.V ; H / D H .
Proof. (i) First of all let us notice that
C
.V ; H / H
since C .V ; H / is orthogonal to V , as it follows from the fact that V is null. Let
V0 WD C .H; H / V and let V1 be the orthogonal complement of V0 in V . From the
definition of V1 it follows that C .H; V1 / is orthogonal to H thus C .H; V1 / D 0.
But C .V ; V1 / D 0 as well since V is null, in other words C .V1 ; / D 0 and we
conclude that V1 D 0 using that C is nondegenerate. This proves (i).
(ii) is proved by an argument similar to that in (i), which we leave to the reader. We end this section with the following:
Definition 5.4. An SNK-holonomy system .V 2m ; g; J;
C
; R/ is said to split as
V D V1 ˚ V2
if V admits an h-invariant, orthogonal and J -invariant splitting V D V1 ˚ V2 such
that
C
belongs to 3 .V1 / ˚ 3 .V2 /:
It is clear that if an SNK-holonomy system splits as V D V1 ˚ V2 then each of
C
/, k D 1; 2 is again an SNK-holonomy system. Note that factors
.Vk ; gjVk ; JjVk ; jV
k
of dimension 4 are not permitted by this definition since in dimension 2 or 4 there
are no non-zero holomorphic 3-forms. Also note that this type of decomposition of
an SNK-holonomy system corresponds exactly to local products of SNK-manifolds.
5.2 The structure of the form
C
We are now ready to have a look at the structure of the form C when a complex
holonomy reduction is given. The starting point of our approach to the classification
problem of reducible SNK-holonomy systems is:
Proposition 5.1. Let V be a proper, J -invariant subspace of .h; V /. If H denotes the
orthogonal complement of V in V , the following hold:
(i) .
C
x
B
C
v /w
(ii) .
C
x
B
C
y /z
belongs to H whenever x, y, z are in H ;
(iii) .
C
v
B
C
w /x
belongs to H for all x in H and v, w in V;
(iv) .
C
x
B
C
y /v
is in V, for all x, y in H and all v in V .
D 0 for all x in H and v, w in V;
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
39
Proof. Directly from the first Bianchi identity for R, combined with the fact that
R.V; H / D 0 we get
R.x; y; v; w/ D hŒ
C
v ;
C
w x; yi
h
C
v w;
C
x ; yi
(5.2)
for all x in H; y in V and v, w in V. All claims are now easy consequences of the
fact that R belongs to S 2 .1;1 / and C is in 3 , see [49] for details.
We can show that any complex reducible SNK-holonomy system contains, up to
products, a null invariant sub-space.
Proposition 5.2. Let .V 2m ; g;
Then V splits as
C
; R/ be a complex reducible SNK-holonomy system.
V D V1 ˚ V2 ;
where V2 contains a null invariant space.
Proof. The proof is completed in two steps we shall outline below.
Step 1: Existence of an isotropic invariant subspace.
The reducibility of .h; V / implies the existence of an invariant splitting
V DE ˚F
which is moreover orthogonal and stable under J . Let F0 be the subspace of F spanned
by f. vC w/F W v; w in Eg, where the subscript indicates orthogonal projection. Using
Proposition 5.1 (i) we get
C
x
C
y ..
C
v w/F /
C
C
x
C
y ..
C
v w/E /
D0
for all x, y in F and whenever v, w are in E. But the first summand is in E by
Proposition 5.1 (iv) while the second is in F by (ii) of the same proposition. therefore
both summands vanish individually, and a positivity argument yields then
C
.F; F0 / D 0:
In particular F0 is null and from (5.1) and the h-invariance of E and F we also get
that F0 is h-invariant. Now F0 ¤ V since C ¤ 0 and if F0 D 0 we have that E is
isotropic, that is C .E; E/ E.
Step 2: Existence of an invariant nullspace.
Using Step 1, we can find a h-invariant subspace , say V in V , which is isotropic in
the sense that C .V ; V / V . Let us consider now the h-invariant tensor r1 W V ! V
given by
hr1 v; wi D T rV . vC B wC /
for all v, w in V . Obviously, this is symmetric and J -invariant. It follows that there
is an orthogonal, h-invariant splitting
V D V0 ˚ V1 ;
in the display, is this
T r
or
Tr for trace?
40
Paul-Andi Nagy
P
where V0 D Ker.r1 /. Since V is isotropic, r1 is given by r1 D dkD1 . vCk /2 v for
all v in V , where d WD dimR V and fvk ; 1 k d g is an orthonormal basis in V.
Then Proposition 5.1 (i) implies that xC .r1 v/ D 0 for all x in H and for all v in V,
in other words
C
.H; V1 / D 0:
But the definition of V0 yields that C .V ; V0 / D 0, in particular V0 is null. We now
form the h and J -invariant subspace H1 D V0 ˚ H0 which is easily seen to satisfy
that
C
.H1 ; H1 / H1 ; C .V1 ; V1 / V1 ; C .V1 ; H1 / D 0:
In other words V D V1 ˚ H1 is a splitting of our holonomy system in the sense of
Definition 5.4 and the result is proved.
This can be furthermore (see [49] for details) refined to:
Proposition 5.3. Let .V 2m ; g; J; C ; R/ be an SNK-holonomy system. If it contains
a proper invariant nullspace, it splits as
V D V1 ˚ V2 ;
where V2 contains a special invariant subspace.
Summarising the results obtained up to now we obtain, after an easy induction
argument on the irreducible components of .h; V /:
Theorem 5.1. Let .V 2m ; g; J; C ; R/ be a complex reducible SNK-holonomy system.
Then V is product of SNK-holonomy systems belonging to one of the following classes:
(i) irreducible SNK-holonomy systems;
(ii) SNK-holonomy systems which contain a special invariant subspace.
To advance with the classification of our holonomy systems we need therefore
only to discuss the second class present in the theorem above. We first observe that
given a special invariant subspace in some SNK-holonomy system one can explicitly
determine the curvature along the special subspace.
Proposition 5.4. Let .V 2m ; g; J; C ; R/ be an SNK-holonomy system containing a
special invariant subspace V . Then:
R.
C
x y; v1 ; v2 ; v3 /
D hŒ
C
v1 ; Œ
C
v2 ;
C
v3 x; Jyi
whenever x, y are in H D V ? and for all vk , 1 k 3 in V .
Actually, Proposition 5.4 singles out a second Lie algebra of relevance for us,
obtained as follows. Consider the subspace p 2 .H / given by
p WD f
together with the subspace q of 1;1
C
v
W v in Vg
.H / given by q WD Œp; p.
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
41
Proposition 5.5. The following hold:
(i) p \ q D 0;
(ii) r D p ˚ q is a Lie subalgebra of ƒ2 .H /.
Proof. From Proposition 5.4 we get
Œ
C
v1 ; Œ
C
v2 ;
C
v3 D
C
R.v1 ;v2 /v3
for all vk ; 1 k 3 in V . That is p is a Lie triple system in the sense that Œp; Œp; p p
hence p C q is a Lie algebra (see [36] for details). To prove (i), let us pick z in
p \ q. Then z D vC for some v in V and moreover Œz; p p. In particular
ΠvC ; JCv D wC for some w in V , or equivalently 2. vC /2 D JCw . But the lefthand side of this equality is symmetric whilst the right-hand is skew-symmetric which
yields easily that v D 0.
The Lie algebra r is best though of as the holonomy algebra of a symmetric space
of compact type. In the realm of NK-geometry this will turn out to be precisely the
case.
5.3 Reduction to a Riemannian holonomy system
We shall consider in what follows an SNK-holonomy system .V 2m ; g; J; C ; R/
containing a special invariant sub-space V , with orthogonal complement to be denoted
by H .
At this moment need a finer notion of irreducibility for an SNK-holonomy system
containing an h-invariant special subspace. Let us now define RH W ƒ2 .H / ! ƒ2 .H /
by
RH .x; y/ D R.x; y/ C CC
x
y
for all x, y in H . Note this is well defined because C .H; H / V and C .V ; H / D
H and also because H is h-invariant. Moreover, the Bianchi identity for R ensures
that RH is an algebraic curvature tensor on H , that is RH belongs to K.so.H //. The
second Lie subalgebra of ƒ2 .H / of interest for us is
hH D LiefRH .x; y/ W x; y in H g:
Definition 5.5. .H; g; hH / is called the Riemannian holonomy system associated to
.V 2m ; g; J; C ; R/. It is irreducible if the metric representation .hH ; H / is irreducible.
The main result in this section is to prove that we can reduce, up to products in
the sense of Definition 5.4 – hence of Riemannian type, the study of special SNKholonomy to the case when the associated Riemannian holonomy system is irreducible.
42
Paul-Andi Nagy
Theorem 5.2. Let .V 2m ; g; J; C ; R/ be an SNK-holonomy system containing an
h-invariant subspace V . Then V splits as
V D V1 ˚ ˚ Vq ;
a product of special SNK-holonomy systems, such that each factor has irreducible
associated Riemannian holonomy system.
Proof. Let V be a special h-invariant subspace and let us orthogonally decompose
H D H1 ˚ H 2
in invariant subspaces for the action of hH . As a straightforward consequence of
having R in S 2 .1;1 / we note that
RH .
C
v /
D
1
2
C
rv
(5.3)
on H , for all v in V . Since RH .H; H; H1 ; H2 / D 0 by assumption and since r is
invertible on V it follows that VC H1 and H2 are orthogonal, hence
C
V Hk
Hk ;
k D 1; 2
given that VC H H . It also follows that
several preliminary steps.
C
(5.4)
.H1 ; H2 / D 0. We will need now
Step 1: Hk are J -invariant for k D 1; 2.
From (5.4) it follows that r V .Hk / Hk , k D 1; 2 where r V W H ! H is defined
by
rVx D d
X
.
C 2
vk / x
kD1
for all x in H , where fvk ; 1 k d D d i mR Vg is some orthonormal basis in V.
But r V has no kernel, as an easy consequence of the fact that C is non-degenerate
V
and V is special. It follows that rjH
W Hk ! Hk is injective, hence surjective, that is
k
r V .Hk / D Hk
for k D 1; 2. Again from (5.4) we find that r V .JHk / Hk hence after applying
.r V /1 (which as we have seen preserves Hk ) we get that JHk Hk for k D 1; 2.
Step 2: A first decomposition of V .
Let V 0 WD V ˚ H1 and consider the orthogonal and J -invariant splitting
V D V 0 ˚ H2 :
Since RH .H1 ; H2 ; H; H / D 0 and also because C .H1 ; H2 / D 0 we find that
R.H1 ; H2 ; H; H / D 0. In other words we have R.H; H /H2 H2 and also
R.H; V /H2 D 0 since V is h-invariant and R is symmetric. Now the Bianchi identity
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
43
for R, combined again with the h-invariance of V and its nullity gives
R.v; w/x D Œ
C
v ;
C
w x
for all v, w in V and for all x in H . Using again (5.4) we find that R.V; V/H2 H2
so H2 is actually h-invariant.
Step 3: The decomposition of V.
Thus we may apply Proposition 5.1 (i) to the splitting V D V 0 ˚ H2 whence
C
x
C
v w
D0
for all x in H2 and whenever v, w belong to V 0 . Let us define Vk D
for k D 1; 2 and notice these are J -invariant. Then
C
C
.Hk ; Hk / V
.H2 ; V 1 / D 0
in particular V 1 and V 2 are orthogonal, after taking the scalar product with elements
in H2 . Therefore
V D V1 ˚ V2
after using that C .H; H / D V and C .H1 ; H2 / D 0.
Now since H1 and H2 play equal rôles, by repeating the arguments above we also
get that C .V 2 ; H1 / D 0.
Step 4: Proof of the theorem.
We consider the orthogonal and J -invariant splitting
V D V1 ˚ V2 ;
where Vk WD V k ˚ Hk , k D 1; 2. From Step 3 it follows that
C
belongs to
.V1 / ˚ .V2 /
3
3
and moreover that Hk are invariant under h for k D 1; 2. Let us prove that, say V 1 ,
is h-invariant too. From (5.1) we get
R.x; y/.
C
x 1 x2 /
D
C
x
R.x;y/x1 2
C
C
x1 R.x; y/x2
for all x, y in V and x1 ; x2 in H1 and for all x, y in H . Since H is h-invariant and we
have seen that C .H; H1 / D C .H1 ; H1 / D V 1 , the invariance of of V 1 follows
and that of V 2 is proved analogously. We conclude that V k are invariant under h, for
k D 1; 2.
The claim follows now by induction on the number of irreducible components of
.hH ; H /, provided we show that hH does not fix any vector in H . But this is implied
by (5.3), for such a vector, say x0 in H would satisfy then C .V ; x0 / D 0. Taking
scalar products with elements in H this yields C .x0 ; H / D 0 as C .H; H / V .
This is to say that xC0 D 0 hence x0 D 0 since C is non-degenerate. This proves the
absence of fixed points for the representation .hH ; H / and the proof is finished. More properties of the non-Riemannian holonomy representation .h; V / can now
be formulated.
44
Paul-Andi Nagy
Proposition 5.6. Let .V 2m ; g; J; C / be a special SNK-holonomy system containing
an invariant subspace V and such that the Riemannian holonomy system .hH ; H / is
irreducible. Then the representation .h; V / is irreducible over C.
Proof. Let us suppose that V is not irreducible and split V D V 1 ˚ V 2 as orthogonal
sum of J -stable and h-invariant subspaces. Then R. xC y; v; v1 ; v2 / D 0 for all x, y
in H , v in V and for all vk in V k , k D 1; 2. Using Proposition 5.4 this leads to
Œ
C
v ;Œ
C
v1 ;
C
v2 D0
whenever v belongs to V and v1 , v2 are in V 1 , V 2 respectively. An easy invariance
argument which can be found in [49], page 492 yields
C
v1
C
v2
D0
(5.5)
for all vk in V k , k D 1; 2. We form H k WD V k H , k D 1; 2 which are therefore
orthogonal, J -invariant and such that H D H 1 ˚ H 2 . The spaces H k , k D 1; 2 are
actually invariant under h a fact which follows from (5.1) and the h-invariance of V k ,
k D 1; 2. Now by using (5.5) we obtain that
C
C
x y
Hk Hk
for all x, y in H , since VCk H H and C .H; H / V. This means, see also
the definition of the tensor RH , that H k , k D 1; 2 are invariant under hH hence they
cannot be both proper since .hH ; H / is irreducible. If H 1 D 0 for instance, a routine
argument leads to V 1 D 0, a contradiction and the proof is finished.
5.4 Metric properties
Let .V; g; J; C ; R/ be an SNK-holonomy system with special invariant subspace
V and such that the associated representation .hH ; H / is irreducible. Let us define
the tensors Ric, Ric and C exactly as in the beginning of this section, and note they
are still subject to (4.4) and (4.5) as it follows from the corresponding proofs. Their
invariance under h is an easy consequence of (4.4). We shall prove here that one
reduces the discussion to the case when C has only a few eigenvalues but skip most
of the technical details. The eigenspaces of the tensor C will be used to construct
invariant subspaces of the representation .h; H / which is not necessarily irreducible,
in contrast to .h; V / which is irreducible by Proposition 5.6.
Proposition 5.7. The tensor C has at most 3-eigenvalues.
Proof. Directly from its definition C preserves the splitting V D V ˚ H . Since
C is h-invariant and .h; V / is irreducible it follows that CjV D 1V for some real
number . Using (4.5) we find that the symmetric and J -invariant tensor S W H ! H ,
S WD CjH C 2 satisfies
S. vC x/ D vC S x;
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
hence
S 2.
C
v x/
C 2
v S x
D
45
(5.6)
for all x in H and for all v in V. Since C is h-invariant, so is S , and in fact the latter
turns out to be hH -invariant, after using (5.6). It follows that S 2 is a multiple of the
identity and the claim follows.
2
Let us consider the tensor r V W H ! H given by
rV D 2d
X
.
C 2
vk / ;
kD1
where 2d D dimR V and fvk ; 1 k 2d g is some orthonormal basis in V . It is easy
to see that r V is positive, J -commuting, symmetric and h-invariant and moreover that
rjH D 2r V .
When C has only one eigenvalue, then C D 0 and one can show [49] by using
(4.4) that both Ric and r must actually by diagonal on V .
It remains to investigate the cases when C has two, respectively three eigenvalues
which are described below.
Proposition 5.8. The following hold:
(i) If C has exactly two eigenvalues then there exists k > 0 such that r V D k1H
and moreover the eigenvalues together with the corresponding eigenspaces for
the tensors r, C , Ric are given in the following table:
Eigenvalue
r
Ric
C
Eigenspace
1
nd
k
d
nC7d
k
4d
4.n3d /
k
d
V
2
2k
nC2d
k
2d
/
2.n3d
k
d
H
(ii) If C has exactly 3-eigenvalues we have a h-invariant, orthogonal and J -invariant splitting
V D V ˚ H 1 ˚ H2
such that
C
.H1 ; H2 / D
C
.H2 ; H2 / D 0 and
C
.H1 ; H2 / D V.
(iii) In all cases, including when C D 0, we have
RicH D gjH
for some > 0, where RicH denotes the Ricci contraction of RH .
We refer the reader to [49] for details of the proof and also mention that in case (ii)
above it is also possible to display the relations between the eigenvalues of all relevant
symmetric endomorphisms.
46
Paul-Andi Nagy
5.5 The final classification
By Theorem 5.2 it is enough to consider an SNK-holonomy system .V 2m ; g; J; C ; R/
with a special invariant subspace V and such that the Riemannian holonomy system
.hH ; H / is irreducible, in the notations of the previous section. We shall combine
here some well-known results for irreducible Riemannian holonomy systems and the
information which is available in our present context. To progress in this direction we
will first make clear the relationship between the Lie algebra r and the Riemannian
holonomy algebra hH .
Proposition 5.9. The following hold:
(i) r is an ideal in hH ;
(ii) RH D 2 1r on r for some > 0 such that rjV D 1V .
Proof. (i) That p hH follows from (5.3) hence q D Œp; p is contained in hH as
well. To see that r is an ideal we use (5.1) to observe that
ŒR.x; y/;
C
v belongs to p for all x, y in H and for all v in V. The definition of RH and that of r
now yields
ŒRH .x; y/; vC in r and it easy to conclude by using the Jacobi identity and Proposition 5.5.
(ii) First of all let us observe that r preserves V , since the latter is special. The
h-invariance of r, together with the irreducibility of .h; V/ implies the existence of a
constant > 0 such that rjV D 1V . For the action of RH on p our claim follows
now from (5.3). To prove it on q we observe we proceed as follows. Given that R
belongs to S 2 .ƒ2 / we can alternatively rewrite (5.1) under the form
R.
C
x1 x2 ; x3 /
C R.x2 ;
C
x1 x3 /
D R.x1 ;
C
x2 x3 /
whenever xk are in V , k D 1; 2; 3. Using this for x1 D v; x2 D wC ei ; x3 D ei where
v, w are in V and fei g is some orthonormal basis in H yields after summation and
rearrangement of terms
X
R.ei ; ΠvC ; wC ei / D R.v; rw/
i
that is R.ΠvC ; wC / D 2 ΠvC ; wC on H , after also using (5.2). After a straightforward invariance argument this leads to RH .ΠvC ; wC / D 2 ΠvC ; wC and the claim
is proved.
Now we recall that for a given metric representation .l; W / of some Lie algebra l
we have the Berger list of irreducible Riemannian holonomies, and proceed to the
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
47
final classification result, hence proving Theorem 1.1 in the introduction.
l
W
so.n/
Rn
u.m/
R2m
su.m/
R2m
sp.m/ ˚ sp.1/
R4m
sp.m/
R4m
spin.7/
R8
g2
R7
Theorem 5.3. Let .M 2m ; g; J / be a complete SNK-manifold. Then M is, up to finite
cover, a Riemannian product whose factors belong to the following classes:
(i) homogeneous SNK-manifolds;
(ii) twistor spaces over positive quaternionic Kähler manifolds;
(iii) 6-dimensional SNK-manifolds.
Proof. First of all, because M must be compact with finite fundamental group we
may assume up to a finite cover that it is simply connected. Pick now a point x in
x Tx M /. If this is irreducible
M and consider the holonomy representation .Holx .r/;
over R or complex irreducible but real reducible we conclude by Theorem 4.4.
x is reducible over C at x we consider the SNKIf the holonomy representation of r
C x
holonomy system .Tx M; gx ; Jx ; x ; Rx /. By the theorem of Ambrose–Singer we
x Also note that when starting the whole reduction procedure
have that h holx .r/.
x complex invariant sub-space we end up
leading to Theorem 5.2 from some holx .r/
with a splitting
Tx M D V1 ˚ ˚ Vp
x
with the properties in Theorem 5.2 and which is furthermore holx .r/-invariant
together with the special sub-spaces contained in each factor. Using parallel transport
x this extends to a r-parallel
x
w.r.t. to r
decomposition, which is actually r-parallel
x is split along the decomposition. The splitting theorem of de
since the torsion of r
Rham applies and our study is reduced to that of SNK-manifolds such that at each
point, we have an invariant special subspace having the property that the associated
holonomy system is irreducible in the sense of Definition 5.5. Then we show (see
[49]) that M is the total space of a Riemannian submersion
W .M; g/ ! .N 2n ; h/
cancel
empty
column or is
something
missing?
48
Paul-Andi Nagy
whose fibres are totally geodesic and w.r.t. the induced metric and almost complex
structure are simply connected, irreducible and compact Hermitian symmetric spaces.
The tensor RH defined in Section 5.3 projects onto the Riemann curvature tensor of
h making of hH a subalgebra of hol.N; h/ at each point. Using this, one shows that
the Riemannian manifold .N 2n ; h/ is irreducible and Proposition 5.8 (iii) gives that h
is Einstein of positive scalar curvature.
If the tensor C has three eigenvalues a short manipulation of the second Bianchi
x combined with the algebraic structure of the splitting of TM given in
identity for r
x is an Ambrose–Singer connection.
(ii) of Proposition 5.8 gives that r
Let us discuss now the case when C has two eigenvalues or it vanishes.
If .N 2n ; h/ is a symmetric space the relations of O’Neill (see [6]) for the Riex is an
mannian submersion W M ! N essentially say that the curvature tensor R
C
H
; R and the curvature of the fibre (which is a symmetric
explicit expression of
x
space) along the r-parallel
splitting TM D V ˚ H . This information results easily
x an Ambrose–Singer connection again.
in having r
Suppose now that .N 2n ; h/ is not a symmetric space. Then the holonomy representation of .N; h/, at the Lie algebra level, corresponds to the representation .hH ; H /
and as it is well known (see [52]), must be one of the entries in the Berger list above. We
shall mainly use now Proposition 5.9. The Lie algebras su.m/, sp.m/, spin.7/, g2 are
excluded because their curvature tensors are Ricci-flat. Supposing that hH D u.n/,
given that the ideal r is at least two-dimensional we can only have r D su.n/; u.n/.
But using Proposition 5.9 combined with the fact that a curvature tensor of KählerEinstein type is completely determined by its restriction to su.m/ we find that .N 2n ; h/
is symmetric a contradiction. Similar arguments together with results from [5] enable
us to conclude in the case when hH D so.2n/ or when r D sp.m/. The only case
remaining is when hH D sp.m/ ˚ sp.1/ and r D sp.1/ which implies that V is of
x is contained in U.1/ U.2m/ and the fact
rank 2. Hence the holonomy group of r
2m
that .M ; g; J / is a twistor space over a positive quaternionic-Kähler manifold has
been proved in [48].
6 Concluding remarks
Due to considerations of space and time we have omitted to present some important
facts related to nearly-Kähler geometry. The first aspect is related to the construction
of examples, which is implicit in our present treatment. Given a quaternion-Kähler
manifold .M 4m ; g; Q/ of positive scalar curvature the Salamon twistor construction
[51] yields a Kähler manifold .Z; h; J / such that there is a Riemannian submersion
with totally geodesic, complex fibres
S 2 ,! .Z; h/ ! .M; g/:
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
49
Using this structure and a canonical variation of the metric h it has been shown [19] that
Z admits an SNK-metric. Enlarging the context to that of Riemannian submersions
with complex, totally geodesic fibres from Kähler manifolds one draws a similar
conclusion [48] (see also [50] for some related facts). This observation can also be
used to describe the homogeneous SNK-manifolds which appear in Theorem 5.3 as
twistor spaces in the sense of [11] over symmetric spaces of compact type.
We have also omitted to discuss NK-structures in dimension 6, which have a rich
geometry, although not yet fully understood. If .M 6 ; g; J / is a strict NK manifold the
structure group automatically reduces to SU.3/, hence c1 .M; J / D 0 [33]. Moreover
the metric g must be Einstein [33], of positive scalar curvature. There is also a 1-1
correspondence between 6-dimensional manifolds admitting real Killing spinors and
SNK-structures in dimension 6 [35].
In the compact case the only known examples are homogeneous, and various characterisations of these instances are available [12], [45]. However, compact examples
with two conical singularities have been very recently constructed in [23] building
on the fact that one can suitably rotate the SU.2/-structure of a Sasakian–Einstein
5-manifold and then extend it to a SU.3/-structure of NK-type through a generalised
cone construction.
Acknowledgments. Part of this material was elaborated during the “77ème rencontre
entre physiciens théoriciens et mathématiciens” held in Strasbourg in 2005. Warm
thanks to the organisers and especially to V.Cortès for a beautiful and inspiring conference. I am also grateful to V.Cortès and A.Moroianu for useful comments.
References
[1]
I. Agricola, Th. Friedrich, P.-A.Nagy, and C. Puhle, On the Ricci tensor in type II B string
theory. Classical Quantum Gravity 22 (2005), no. 13, 2569–2577.
[2]
B. Alexandrov, Sp.n/U.1/-connections with parallel totally skew-symmetric torsion.
J. Geom. Phys. 57 (2006), no. 1, 323–337.
[3]
B. Alexandrov, On weak holonomy. Math. Scand. 96 (2005), no. 2, 169–189.
[4]
B. Alexandrov, Th. Friedrich, and N. Schoemann, Almost Hermitian 6-manifolds revisited.
J. Geom. Phys. 53 (2005), no. 1, 1–30.
[5]
F. Belgun and A. Moroianu, Nearly-Kähler 6-manifolds with reduced holonomy. Ann.
Global Anal. Geom. 19 (2001), 307–319.
[6]
A. Besse, Einstein manifolds. Ergeb. Math. Grenzgeb. 10, Springer-Verlag, Berlin 1987.
[7]
J.-M. Bismut, A local index theorem for non-Kähler manifolds. Math. Ann. 284 (1989),
681–699.
[8]
M. Bobieński and P. Nurowski, Irreducible SO.3/ geometry in dimension five. J. Reine
Angew. Math. 605 (2007), 51–93.
50
[9]
Paul-Andi Nagy
R. L. Bryant, Classical, exceptional, and exotic holonomies: a status report. In Actes de
la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr. 1, Soc. Math.
France, Paris 1996, 93–165.
[10] R. Bryant, Recent advances in the theory of holonomy. Astérisque (2000), Exp. No. 861,
5, 351–374, séminaire Bourbaki, Vol. 1998/99.
[11] F. E. Burstall and J. H. Rawnley, Twistor theory for Riemannian symmetric spaces. Lecture
Notes in Math. 1424, Springer-Verlag, Berlin 1990
[12] J.-B. Butruille, Classification des variétés approximativement kähleriennes homogènes.
Ann. Global Anal. Geom. 27 (2005), 201–225.
[13] J.-B. Butruille, Espace de twisteurs d’une variété presque hermitienne de dimension 6.
Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1451–1485.
[14] S. Chiossi and S. Salamon, The intrinsic torsion of SU.3/ and G2 structures. In Differential
geometry, Valencia, 2001, World Scientific Publishing, River Edge, NJ, 2002, 115–133.
[15] R. Cleyton and S. Ivanov, Curvature decomposition of G2 manifolds. J. Geom. Phys. 58
(2008), no. 10, 1429–1449.
[16] R. Cleyton and S. Ivanov, Conformal equivalence between certain geometries in dimension
6 and 7. Math. Res. Lett. 15 (2008), no. 4, 631–640.
[17] R. Cleyton and A. Swann, Einstein metrics via intrinsic or parallel torsion. Math. Z. 247
(2004), 513–528.
[18] V. Cortés and L. Schäfer, Flat nearly Kähler manifolds. Ann. Global Anal. Geom. 32
(2007), 379–389.
[19] J. Eells and S. Salamon, Constructions twistorielles des applications harmoniques. C. R.
Acad. Sci. Paris Sér. I Math. 296 (1983), 685–687.
[20] M. Falcitelli,A. Farinola, and S. Salamon,Almost-Hermitian geometry. Differential Geom.
Appl. 4 (1994), 259–282.
[21] M. Fernández, A classification of Riemannian manifolds with structure group Spin.7/.
Ann. Mat. Pura Appl. (4) 143 (1986), 101–122.
[22] M. Fernández and A. Gray, Riemannian manifolds with structure group G2 . Ann. Mat.
Pura Appl. (4) 132 (1982), 19–45 (1983).
[23] M. Fernández, S. Ivanov, V. Muñoz, and L. Ugarte, Nearly hypo structures and compact
nearly Kähler 6-manifolds with conical singularities. J. London Math. Soc. (2) 78 (2008),
no. 3, 580–604.
[24] T. Friedrich, Weak Spin(9)-structures on 16-dimensional Riemannian manifolds. Asian
J. Math. 5 (2001), 129–160.
[25] T. Friedrich, On types of non-integrable geometries. Rend. Circ. Mat. Palermo (2) Suppl.
No. 71 (2003), 99–113.
[26] T. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion
in string theory. Asian J. Math. 6 (2002), 303–335.
[27] T. Friedrich, I. Kath, A. Moroianu, and U. Semmelmann, On nearly parallel G2 -structures.
J. Geom. Phys. 23 (1997), 259–286.
[28] P. Gauduchon, Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B (7) 11
(1997), 257–288.
Chapter 10. Totally skew-symmetric torsion and nearly-Kähler geometry
51
[29] G. Grantcharov and Y. S. Poon, Geometry of hyper-Kähler connections with torsion.
Comm. Math. Phys. 213 (2000), 19–37.
[30] A. Gray, Nearly Kähler manifolds. J. Differential Geometry 4 (1970), 283–309.
[31] A. Gray, Weak holonomy groups. Math. Z. 123 (1971), 290–300.
[32] A. Gray, Riemannian manifolds with geodesic symmetries of order 3. J. Differential Geometry 7 (1972), 343–369.
[33] A. Gray, The structure of nearly Kähler manifolds. Math. Ann. 223 (1976), 233–248.
[34] A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their
linear invariants. Ann. Mat. Pura Appl. (4) 123 (1980), 35–58.
[35] R. Grunewald, Six-dimensional Riemannian manifolds with a real Killing spinor. Ann.
Global Anal. Geom. 8 (1990), 43–59.
[36] S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Graduate Studies in
Mathematics 34, American Mathematical Society, Providence, RI 2001, corrected reprint
of the 1978 original.
[37] N. Hitchin, Stable forms and special metrics. In Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math. 288, Amer. Math. Soc.,
Providence, RI 2001, 70–89.
[38] P. S. Howe and G. Papadopoulos, Twistor spaces for hyper-Kähler manifolds with torsion.
Phys. Lett. B 379 (1996), 80–86.
[39] S. Ivanov, Geometry of quaternionic Kähler connections with torsion. J. Geom. Phys. 41
(2002), 235–257.
[40] V. F. Kiričenko, K-spaces of maximal rank. Mat. Zametki 22 (1977), 465–476.
[41] F. Martín Cabrera and A. Swann, Almost Hermitian structures and quaternionic geometries. Differential Geom. Appl. 21 (2004), 199–214.
[42] F. Martín Cabrera and A. Swann, Curvature of special almost Hermitian manifolds. Pacific
J. Math. 228 (2006), 165–184.
[43] F. Martín Cabrera and A. Swann, The intrinsic torsion of almost quaternion-Hermitian
manifolds. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 5, 1455–1497.
[44] S. Merkulov and L. Schwachhöfer, Classification of irreducible holonomies of torsion-free
connections. Ann. of Math. (2) 150 (1999), no.1, 77–149.
[45] A. Moroianu, P.-A. Nagy, and U. Semmelmann, Unit Killing vector fields on nearly Kähler
manifolds. Internat. J. Math. 16 (2005), 281–301.
[46] A. Moroianu, P.-A. Nagy, and U. Semmelmann, Deformations of nearly-Kähler structures.
Pacific J. Math. 235 (2008), no. 1, 57-72
[47] A. Moroianu and L. Ornea, Conformally Einstein products and nearly Kähler manifolds.
Ann. Global Anal. Geom. 33 (2008), no. 1, 11–18.
[48] P.-A. Nagy, On nearly-Kähler geometry. Ann. Global Anal. Geom. 22 (2002), 167–178.
[49] P.-A. Nagy, Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6 (2002),
481–504.
[50] P.-A. Nagy, Rigidity of Riemannian foliations with complex leaves on Kähler manifolds.
J. Geom. Anal. 13 (2003), 659–667.
52
Paul-Andi Nagy
[51] S. Salamon, Quaternionic Kähler manifolds. Invent. Math. 67 (1982), 143–171.
[52] S. Salamon, Riemannian geometry and holonomy groups. Pitman Research Notes in Math.
Ser. 201, Longman Scientific & Technical, Harlow 1989.
[53] U. Semmelmann, Conformal Killing forms on Riemannian manifolds. Math. Z. 245
(2003), 503–527.
[54] A. Strominger, Superstrings with torsion. Nuclear Phys. B 274 (1986), 253–284.
[55] A. Swann, Weakening holonomy. In Quaternionic structures in mathematics and physics
(Rome, 1999), Università degli Studi di Roma “La Sapienza”, Rome, 1999, 405–415
(http://www.emis.de/proceedings/QSMP99/index.html).
[56] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds. Trans.
Amer. Math. Soc. 267 (1981), 365–397.
[57] F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds. London
Math. Soc. Lecture Note Ser. 83, Cambridge University Press, Cambridge 1983.
[58] F. Witt, Special metrics and triality. Adv. Math. 219 (2008), 1972–2005.
Index
Ambrose–Singer connection, 35
Berger list, 46
Bianchi map, 6
bidegree splitting, 4
characteristic connection, 15
Chern form, 34
Cleyton–Swann theorem, 35
comparison formula, 22
curvature tensor, 6, 26
decomposition of curvature, 9, 26
Gray–Hervella class, 15
G1 -structures, 22
Hermitian connection, 3, 11
Hermitian Killing form, 17
Hermitian Ricci tensor, 33
holonomy representation, 35
holonomy system, 36
intrinsic torsion tensor, 13
isotropic subspace, 37
isotropy algebra, 37
Kähler form, 4
Killing form, 16, 31
Killing vector field, 19
nearly-Kähler, 31
Nijenhuis tensor, 2, 11
null subspace, 37
Riemannian submersion, 47
special subspace, 37, 41
strict nearly-Kähler, 32, 33
torsion tensor, 3, 14, 35
twistor space, 3, 47
W1 C W4 structure, 25
53